Ancient traditions often assume or imply the Mosaic authorship of the Pentateuch:

• The books of Joshua (8:31-32, 23:6) and Kings (1 Kings 2:3; 2 Kings 14:6, 23:25) refer to the torah of Moses, or the scroll of the torah of Moses—though these are likely references to (some form of) Deuteronomy, not the entire Torah in its current form.
• The much later books of Ezra (3:2, 6:18, 7:6) Nehemiah (1:7-9, 8:1, 14, 9:14, 10:30, 13:1), Daniel (9:1, 13), and Chronicles (2 Chron 23:18, 30:16, 34:14) also refer to “the torah of Moses” or paraphrase laws from the Pentateuch as laws of Moses.
• In the New Testament, Luke (2:22) refers to “the law of Moses” and Mark (12:19) states “Moses wrote” followed by a citation of Deut 25:5-6.
• In the Babylonian Talmud, tractate Baba Bathra (14b-15a), Moses is listed as the Pentateuch’s author.

Similar assumptions can be found in other ancient authors such as Philo of Alexandria (d. ca. 50 CE, See, e.g., On the Change of Names, in which he quotes Genesis 1:26 and says “Moses teaches us here by implication the doctrine which he so often lays down that God is the maker of the wise and good only” (4.32, F. H. Colson and G. H. Whitaker’s translation in the Loeb Classical Library, Cambridge: Harvard University Press, 1934, page 159), Josephus (d. ca. 100 CE),[16] and the Early Christian writer Clement of Alexandria (d. ca. 215 CE), For example, he cites the Shema and attributes it to Moses: “the inspired Moses turning us away from all idolatry, utters this truly noble cry: ‘Hear O Israel, the Lord is thy God; the Lord is one” (Exhortation to the Greeks, 8.68, G.W. Butterworth’s translation in the Loeb Classical Library, Cambridge: Harvard University Press, 1979, page 181). Nevertheless, as already noted by Spinoza, these are far from eye-witness accounts. More importantly, these are not statements from the Pentateuch but about it.
Presentation of Moses in the Torah
The Pentateuch does not present itself as being written by Moses, but as an anonymous account about the history of the world and the Israelites up to and including the life of Moses.
No Moses in Genesis:
In fact, the name “Moses” never occurs in Genesis, and nothing in the Torah itself implies that he authored this book. He is first mentioned in Exodus, which records his birth and begins the story of his life. Compare this with how the book of Jubilees presents itself, for instance, in its opening lines:

This is The Account of the Division of Days of the Law… just as the Lord told it to Moses on Mount Sinai when he went up to receive the tablets of the Law and the commandment by the word of the LORD…
This passage explicitly presents Moses as, if not the author, then the transcriber of Jubilees. It also assumes that Moses is the author of the Pentateuch, and that Jubilees is his second book. In contrast, Genesis opens with an anonymous authorial voice describing the creation of the world. Nothing in the biblical book of Genesis is presented as having been “revealed” to Moses; it is simply a series of stories told by an anonymous author. Unlike a reader of Jubilees, the reader of Genesis would have no reason to imagine Moses, or any other named person, as the author (or transcriber) of the book.
Third Person

Furthermore, Moses is referred to about six hundred times in the third person in Exodus, Leviticus, Numbers, and Deuteronomy (“Moses said this,” or “Moses did that”). The name “Moses” is mentioned around 290 times in Exodus, around 85 times in Leviticus, around 230 times in Numbers, and around 35 times in Deuteronomy. It is readily apparent from such statements that someone else is writing about Moses, rather than Moses doing (all) the writing himself.

Moreover, the book of Numbers writes:
וְהָאִישׁ מֹשֶׁה (ענו) [עָנָיו] מְאֹד מִכֹּל הָאָדָם אֲשֶׁר עַל פְּנֵי הָאֲדָמָה.
Now the man Moses was very humble, more so than anyone else on the face of the earth. (Num 12.3)
Not only is this third person narration, but it is certainly not the sort of thing that a humble person would say about him- or herself!
First Person Accounts
Some ancient authors wrote or at least presented their works as first person accounts. The Moabite Mesha inscription is a first person account ostensibly from King Mesha, and the Tel Dan Inscription is a first person account, ostensibly from the Aramean king Hazael (For the text and translation of the Tel Dan Stele, see, Christopher Rollston, “The Tel-Dan Inscription,” Bible Odyssey. For a synopsis of the significance of the Mesha Stele, see Erasmus Gaß, “The Mesha Stela,” Bible Odyssey).
The Bible also has first person accounts, most notably the book of Nehemiah, which is framed (accurately or not) as Nehemiah’s memoir:

נחמיה א:א דִּבְרֵי נְחֶמְיָה בֶּן חֲכַלְיָה וַיְהִי בְחֹדֶשׁ (כסלו) [כִּסְלֵיו] שְׁנַת עֶשְׂרִים וַאֲנִי הָיִיתִי בְּשׁוּשַׁן הַבִּירָה. Neh 1:1 The words of Nehemiah son of Hacaliah. In the month of Chislev, in the twentieth year, while I was in Susa the capital,
Thus, the simplest understanding of the Pentateuch’s own presentation is that someone with (real or perceived) knowledge of the history of the world, and the Israelites in particular, wrote the Pentateuch, and that this person was particularly interested in teaching his readers about Moses and the many messages he believed Moses received from God.
What the Torah Actually Describes Moses Writing Down
The Pentateuch does not refer to Moses as its author, although it refers to Moses writing down select passages.

God’s Promise to the Amalekites – After a battle with the Amalakites,
וַיֹּאמֶר יְהוָה אֶל מֹשֶׁה כְּתֹב זֹאת זִכָּרוֹן בַּסֵּפֶר וְשִׂים בְּאָזְנֵי יְהוֹשֻׁעַ כִי מָחֹה אֶמְחֶה אֶת זֵכֶר עֲמָלֵק מִתַּחַת הַשָּׁמָים. Exod 17:14 Then the LORD said to Moses: “Write this as a reminder in a book and recite it in the hearing of Joshua: ‘I will utterly blot out the remembrance of Amalek from under heaven.’”
Here it seems that Moses was either supposed to write down the sentence, “‘I will utterly blot out the remembrance of Amalek from under heaven,” or a summary of what happened with Amalek, including God’s promised vengeance.

The Covenant Collection – After the core legal section known as the Covenant Collection (Exod 20-23), the narrative says,

שמות כד:ג וַיָּבֹא מֹשֶׁה וַיְסַפֵּר לָעָם אֵת כָּל דִּבְרֵי יְהוָה וְאֵת כָּל הַמִּשְׁפָּטִים… כד:דוַיִּכְתֹּב מֹשֶׁה אֵת כָּל דִּבְרֵי יְהוָה Exod 24:3 Moses came and told the people all the words of the LORD and all the ordinances… 23:4 And Moses wrote down all the words of the LORD…
According to this, Moses wrote down the Covenant Collection on a scroll.

The Decalogue and the Ritual Decalogue – After the destruction of the original stone tablets (Exod 32), God tells Moses to cut two new tablets upon which God will write what was on the former tablets (Exod 34:1-4). Then, after a prayer from Moses (Exod 34:6-9), God makes a covenant again with Israel, including a list of laws (Exod 34:10-26) which scholars refer to as the Ritual Decalogue. The text follows these laws with the following notice:

שמות לד:כז וַיֹּאמֶר יְהוָה אֶל מֹשֶׁה כְּתָב לְךָ אֶת הַדְּבָרִים הָאֵלֶּה כִּי עַל פִּי הַדְּבָרִים הָאֵלֶּה כָּרַתִּי אִתְּךָ בְּרִית וְאֶת יִשְׂרָאֵל. לד:כח וַיְהִי שָׁם עִם יְהוָה אַרְבָּעִים יוֹם וְאַרְבָּעִים לַיְלָה לֶחֶם לֹא אָכַל וּמַיִם לֹא שָׁתָה וַיִּכְתֹּב עַל הַלֻּחֹת אֵת דִּבְרֵי הַבְּרִית עֲשֶׂרֶת הַדְּברים. Exod 34:27 The LORD said to Moses: “Write these words; in accordance with these words I have made a covenant with you and with Israel.” 34:28 He was there with the LORD forty days and forty nights; he neither ate bread nor drank water. And he wrote on the tablets the words of the covenant, the ten commandments.
Although the text is a little hard to follow, it suggests that God writes the same Decalogue again on tablets, whereas Moses writes the new covenant rules, the Ritual Decalogue, perhaps on a scroll.
The List of Stops in the Wilderness – Numbers 33 lists all the places where the Israelites stopped on their way through the wilderness. The chapter begins:

במדבר לג:ב וַיִּכְתֹּב מֹשֶׁה אֶת מוֹצָאֵיהֶם לְמַסְעֵיהֶם עַל פִּי יְהוָה… Num 33:2 Moses wrote down their starting points, stage by stage, by command of the LORD…
According to this, Moses wrote down that list of stops.

Haazinu – Before he dies, Moses teaches the Israelites a song and even wrote it down:

דברים לא:כב וַיִּכְתֹּב מֹשֶׁה אֶת הַשִּׁירָה הַזֹּאת בַּיּוֹם הַהוּא וַיְלַמְּדה אֶת בְּנֵי יִשְׂרָאֵל. Deut 31:22 That very day Moses wrote this song and taught it to the Israelites.
The Core of Deuteronomy – The closest any verse in the Pentateuch comes to stating that Moses wrote the Torah comes towards the end of Deuteronomy, which states:

דברים לא:ט וַיִּכְתֹּב מֹשֶׁה אֶת הַתּוֹרָה הַזֹּאת וַיִּתְּנָהּ אֶל הַכֹּהֲנִים בנֵי לֵוִי הַנֹּשְׂאִים אֶת אֲרוֹן בְּרִית יְהוָה וְאֶל כָּל זִקְנֵי יִשְׂרָאֵל. Deut 31:9 Then Moses wrote down this law, and gave it to the priests, the sons of Levi, who carried the ark of the covenant of the LORD, and to all the elders of Israel.
But to what does “this law” (torah) refer? The context suggests that it refers to the core of Deuteronomy, which is introduced at the beginning of the book as “the torah.”

Deut 1:5
בְּעֵבֶר הַיַּרְדֵּן בְּאֶרֶץ מוֹאָב הוֹאִיל מֹשֶׁה בֵּאֵר אֶת הַתּוֹרָה הַזֹּאת לֵאמֹר. Beyond the Jordan in the land of Moab, Moses undertook to expound this law as follows:
Deut 4:44
וְזֹאת הַתּוֹרָה אֲשֶׁר שָׂם מֹשֶׁה לִפְנֵי בְּנֵי יִשְׂרָאֵל. This is the law that Moses set before the Israelites.
Since chapter 31 is a 3rd person account of what Moses did after he delivered the law (torah) to Israel, including the third person reference to Moses writing the law down, clearly the author of this chapter does not think that what he was writing was part of this torah or on that scroll. This point was made by Menachem Haran. See Menachem Haran, The Biblical Collection vol. 2 (Jerusalem: Magnes, 2003), 66-68 [Hebrew]. Editor’s note: Ibn Ezra seems to have a similar thought in mind as he includes this verse among the verses that Moses could not have written (gloss on Deut 1:1-5).
In other words, the author of Deuteronomy (not Moses) is claiming that included in his book (Deuteronomy or the Pentateuch) is the law that Moses taught to Israel and then wrote down. This is not a claim for Moses writing Deuteronomy, only for much of Deuteronomy coming from a scroll that Moses wrote.
The Torah Uses Moses’ Writings
In short, not only do these texts not claim that Moses wrote the Pentateuch, they actually claim that Moses wrote only certain passages, and even in these narratives, Moses is referred to in the third person. The clear implication is that the author of the Pentateuch, who is emphatically not Moses, is saying that he made use of texts written by Moses, such as the Covenant Collection, the Haazinu Song, etc., and has included them in his book. But he also included texts and traditions that he does not describe as deriving from Moses, such as the quote from the book of the Wars of the Lord (Num 21:13-15), the poem of the balladeers about Heshbon (Num 21:27-30), Lamech’s song to his wives (Gen 4:23-24), and likely many other sources that the author makes use of but does not quote.
Dating the Torah: Long After Moses
Moses’ Death
One glaring example of the post-Mosaic authorship of the Torah is its description of Moses’ death.
דברים לד:ה וַיָּמָת שָׁם מֹשֶׁה עֶבֶד יְהוָה בְּאֶרֶץ מוֹאָב… לד:ז וּמֹשֶׁה בֶּן מֵאָה וְעֶשְׂרִים שָׁנָה בְּמֹתוֹ לֹא כָהֲתָה עֵינוֹ וְלֹא נָס לֵחֹה. Deut 34:5 5 Then Moses, the servant of the LORD, died there in the land of Moab… 34:7 Moses was one hundred twenty years old when he died; his sight was unimpaired and his vigor had not abated.
As already noted by the Talmudic rabbis (b. Baba Batra 14b-15a; Menachot 30a) and the medieval commentator Abraham ibn Ezra, Moses could not have written about his own death.
Why Joshua Cannot Be the Torah’s Author
The Babylonian Talmud (op cit.) records the suggestion that this passage was written by Joshua, but nothing in this passage or in the Pentateuch implies that Joshua wrote it. The rabbis are choosing Joshua because he is Moses’ successor and thus, closest in time. In fact, verse 9 describes Joshua’s actions after Moses’ death in the third person, again implying that someone else is writing.

דברים לד:ט וִיהוֹשֻׁעַ בִּן נוּן מָלֵא רוּחַ חָכְמָה כִּי סָמַךְ מֹשֶׁה אֶת יָדָיו עָלָיו וַיִּשְׁמְעוּ אֵלָיו בְּנֵי יִשְׂרָאֵל וַיַּעֲשׂוּ כאשר צִוָּה יְהוָה אֶת מֹשֶׁה. Deut 34:9 Joshua son of Nun was full of the spirit of wisdom, because Moses had laid his hands on him; and the Israelites obeyed him, doing as the LORD had commanded Moses.
Moreover, the continuation of this passage makes clear that the author cannot be Joshua or anyone who lived at that time:

דברים לד:י וְלֹא קָם נָבִיא עוֹד בְּיִשְׂרָאֵל כְּמֹשֶׁה אֲשֶׁר יְדָעוֹ יְהוָה פָּנִים אֶל פָּנִים. Deut 34:10 Never since has there arisen a prophet in Israel like Moses, whom the LORD knew face to face.
The sentence is in past tense. This evaluation only makes sense if offered by someone living much later than Moses, who can look back and say that since Moses, never has his equal arisen. It would be just as absurd for Joshua to make such a claim as it would be for Moses.
Editorial Comments About Post Wilderness Period Events
In a handful of places, the Pentateuch makes references to matters that show that the author is living in the Cisjordan, long after the wilderness period and the conquest. Many of these were noted by the medieval Jewish commentators R. Abraham ibn Ezra (1089-1167) and R. Judah the Pious (1150-1217). Scholars debate whether this insight comes from him or from his son, Moshe Zaltman, who wrote down his father’s commentary. See discussion in, see Eran Viezel, “R. Judah he-Hasid or R. Moshe Zaltman: Who Proposed that Torah Verses Were Written After the Time of Moses?,” Journal of Jewish Studies 66:1 (2015): 97–115.
Canaanites in the Land – Abram’s arrival in the land is said to be when “the Canaanites were in the land” (Gen 12:6; 13:7). Clearly, the author is living during a time when Canaanites were no longer in the land, yet according to the Bible, the conquest of Canaan occurs after Moses’ death. This example, noted by ibn Ezra, is discussed at length in Eleazer Bonfils’ supercommentary on ibn Ezra (Tzafnat Paneach, ad loc.), see Hebrew-English version here.
On the Lord’s Mountain –
After the binding of Isaac, Abraham names the spot “Adonai-Yireh” (“The Lord will Provide”, Gen 22:14), which is why, the author tells us, “it is said to this day, ‘On the mount of the LORD it shall be provided.’” This refers to a popular ancient Israelite saying about Mount Moriah, which would only make sense after the construction of the Solomonic Temple. This implies that the author of the Pentateuch lived no earlier than the time of Solomon.
First King of Israel – At the end of the description of Esau’s descendants comes the “Edomite King List” which opens with:

בראשית לו:לא וְאֵלֶּה הַמְּלָכִים אֲשֶׁר מָלְכוּ בְּאֶרֶץ אֱדוֹם לִפְנֵי מְלָךְ מֶלֶךְ לִבְנֵי יִשְׂרָאֵל. Gen 36:31 These are the kings who reigned in the land of Edom, before any king reigned over the Israelites.
The first king in Israelite history according to the Bible is Saul (1 Sam 9), and he reigned long after the wilderness period. This again implies that the author must have lived no earlier than the reign of Saul. An unknown Jewish (likely Karaite) interpreter called Yitzhaki noted this example in the 11th century. This is the one case in which ibn Ezra disputes the point. See here for Bonfils’ explanation why. Nevertheless, Judah the Pious (or his son, Moshe Zaltman, who wrote the commentary) agrees with Yitzhaki here; see his gloss on Deut 2:8.
Manna – In Exodus, the account of the manna falling ends with the following statement:

שמות טז:לה וּבְנֵי יִשְׂרָאֵל אָכְלוּ אֶת הַמָּן אַרְבָּעִים שָׁנָה עַד בֹּאָם אֶל אֶרֶץ נוֹשָׁבֶת אֶת הַמָּן אָכְלוּ עַד בֹּאָם אֶל קְצֵה אֶרֶץ כְּנָעַן. Exod 16:35 The Israelites ate manna forty years, until they came to a habitable land; they ate manna, until they came to the border of the land of Canaan.
This verse is written from the vantage point of a writer living after the manna had ceased since it is referring back in time to when it stopped falling. According to Joshua 5:12, this occurred after the Israelites crossed the Jordan River:

יהושע ה:יב וַיִּשְׁבֹּת הַמָּן מִמָּחֳרָת בְּאָכְלָם מֵעֲבוּר הָאָרֶץ וְלֹא הָיָה עוֹד לִבְנֵי יִשְׂרָאֵל מָן וַיֹּאכְלוּ מִתְּבוּאַת אֶרֶץ כְּנַעַן בַּשָּׁנָה הַהִיא. Josh 5:12 The manna ceased on the day they ate the produce of the land, and the Israelites no longer had manna; they ate the crops of the land of Canaan that year.
The Other Side of the Jordan – Deuteronomy begins by describing where Moses and the Israelites were when Moses began to deliver (or write) the speech recorded in Deuteronomy:

דברים א:א אֵלֶּה הַדְּבָרִים אֲשֶׁר דִּבֶּר מֹשֶׁה אֶל כָּל יִשְראֵל בְּעֵבֶר הַיַּרְדֵּן… Deut 1:1 These are the words that Moses spoke to all Israel beyond the Jordan…
If Moses is beyond the Jordan, then the author must not be on that same side (otherwise it wouldn’t be “beyond”). This implies that the author is writing from the Cisjordan, after the Israelite settlement. Ibn Ezra’s gloss on this passage is where he lays out his “secret of the twelve,” namely, that certain verses, like the last 12 verses of the Torah, were not written by Moses. For his full comment and Bonfils’ long discussion of it, see here.
The Conquest that Happened – When describing the history of Mount Seir, Deuteronomy writes:

דברים ב:יב וּבְשֵׂעִיר יָשְׁבוּ הַחֹרִים לְפָנִים וּבְנֵי עֵשָׂו יִירָשׁוּם וַיַּשְׁמִידוּם מִפְּניהֶם וַיֵּשְׁבוּ תַּחְתָּם כַּאֲשֶׁר עָשָׂה ישראל לְאֶרֶץ יְרֻשָּׁתוֹ אֲשֶׁר נָתַן יְהוָה לָהֶם. Deut 2:12 Moreover, the Horim had formerly inhabited Seir, but the descendants of Esau dispossessed them, destroying them and settling in their place, as Israel has done in the land that the LORD gave them as a possession.

The author describes the conquest of the Cisjordan as something that happened in the past; by definition, this must have been written after the settlement period.

Og’s Bed – After describing the conquest of the Bashan, Deuteronomy writes:

דברים ג:יא כִּי רַק עוֹג מֶלֶךְ הַבָּשָׁן נִשְׁאַר מִיֶּתֶר הָרְפָאִים הִנֵּה עַרְשׂוֹ עֶרֶשׂ בַּרזֶל הֲלֹה הִוא בְּרַבַּת בְּנֵי עַמּוֹן… Deut 3:11 Now only King Og of Bashan was left of the remnant of the Rephaim. In fact his bed, an iron bed, can still be seen in Rabbah of the Ammonites…
The text refers to Og’s enormous bed as being in the capital city of Ammon, Rabbah. How did it get there? Would it not have been in Og’s palace in the Bashan, now Israelite territory? This implies that the author is living at a much later time, and that the bed has somehow made its way from the Bashan to Rabbah and is on display there for any who care to see.
The Author of the Torah Continues Beyond the Pentateuch
Reading the Pentateuch as stopping after Deuteronomy is arguably artificial. If it weren’t for the traditional claim that Moses wrote the Torah only, and that the Torah was canonized by Jews (and Samaritans) as separate from the prophetic books, it would certainly be possible to argue that the same anonymous authorial voice continues into Joshua, and perhaps even Judges, Samuel and Kings. In scholarship, this is called the Hexateuch, meaning, the Six Scrolls. See discussion in Marc Brettler, “Is the Torah a Pentateuch or a Hexateuch,” TheTorah.com (2013). This is actually the view of some contemporary scholars, who refer to this whole complex as the Primary History or the Enneateuch (meaning “nine scrolls”).
And so, if we take the Pentateuch seriously, it is clear that all it claims is to be privy to some sources written by Moses, and to knowledge of discourse between Moses and God, or Moses and Israel, just as it does with Abraham, Jacob, Noah, etc. Certain traditions may claim Moses as its author, and thus suggest a 13th century date, but this does not come from the Pentateuch itself; if anything, it flies in the face of the Pentateuch’s self-presentation.

“[Peter :] The Law/Torah was given to Moses orally, not in writing. However, after Moses was taken to heaven, someone else who was not Moses wrote it. After all, how could Moses write about his own death?… It was not the prophets who wrote [the Torah].” | Pseudo-Clementine Homilies 3.47 (4th century)