The Mosaic authorship of the Pentateuch is a view that understands Moses to be the main author of the Pentateuch (i.e. the first five books of the Bible). This belief has received criticism, mostly in the last 200 years, yet remains supported by both Scripture and early church tradition. The conclusion is not that Moses wrote all of the Pentateuch, but that he had a major hand in not only its writing and development, but even its composition (i.e. how it was put together).
Support for Mosaic authorship
Support from the Pentateuch
Throughout the Pentateuch it becomes clear that Moses is recording what happens. Exodus 24:4 states that "Moses then wrote down everything the LORD had said" and that shortly thereafter he took the "Book of the Covenant and read it to the people" (Ex 24:7). Moses also wrote down the Ten Commandments (Ex 24:27-28). Other passages shed light on the fact that Moses kept record of what was going on (Ex 17:14; Num 33:2), and there are clear references that Moses was the author of Deuteronomy (Deut 31:9, 19, 22, 24). Less direct references can be found in Ex 25:16, 21-22; Deut 28:58; 29:20, 21, 27, 29; 20:10, 11.
Support across the Bible
The Bible refers to Moses as the authority behind the books of the Law. The books are referred to as the "Book of Moses" (5 times ), the "Law of Moses" (22 times ), the "Book of the Law of Moses" (4 times ), the "Word of the Lord by Moses" (1 Time ), and the contents of the books are attributed to Moses over 32 times (cf. 2 K. 21:8, 1 Ch. 15:15, 1 Ch. 22:13, 2 Ch 24:6, 2 Ch. 33:8, 2 Ch. 34:14, Ne. 1:8, Ne. 8:14, Ne. 10:29, Mal. 4:4, Matt. 8:4, Matt. 19:8, Matt. 22:14, Mark 1:44, Mark 7:10, Mark 10:4, Luke 5:14, Luke 20:37, Luke 24:27, John 1:45, John 5:46, John 7:19, John 7:22, John 8:5, Acts 3:22, Acts 15:21, Acts 26:22, Rom. 10:5, Rom. 10:19, 2 Cor. 3:15, Heb. 9:19).
Jewish and early Christian support
The Jewish talmud refers to the first five books of the Bible as the "Book of Moses." Furthermore, the Mishna and the early Jewish historian Josephus both accepted the Mosaic authorship of the Pentateuch (Arnold and Beyer, p. 69).
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Support from Jesus
Jesus divided the Old Testament into three sections in Luke 24:27, 44: Moses, the Prophets, and the Psalms. Also, in Mark 10:4-8, Jesus quoted Gen. 2:24 as coming from Moses. In Mark 7:10, Jesus quoted the Ten Commandments as coming from Moses. In Mark 10:3 Jesus refers to Deut. 24:1f as being from Moses, and in Matt. 8:4 Jesus quoted Lev. 14 as coming from Moses. 
Did Moses write everything?
"Although biblical scholarship is deeply divided on the issue of how the Pentateuch was composed, there is widespread agreement that the Pentateuch, as it now stands, is an edited work and not a piece of literature that was penned... by one individual," (Alexander, p. 61-62). This conclusion stems from the fact that the Pentateuch occasionally refers to other pre-existing documents that were written down prior to the writing of Genesis through Deuteronomy (e.g. Gen 5:1; Ex 17:14, 24:7, 34:27; Num 21:14-15, 33:2; Deut 31:9, 22, 24). The idea that Moses wrote all of the Pentateuch also becomes difficult when it is realized that he could not have described his own death and burial in Deut 34.
"While the long-standing tradition of Mosaic authorship is based upon clear statements that Moses was responsible for writing substantial parts of the Pentateuch, the weight of evidence suggests that Moses probably did not compose the Pentateuch as we now have it... This is not to say that the Pentateuch's claims concerning Moses' literary activity should be rejected. On the contrary, such assertions ought to be respected and given serious consideration, which unfortunately all too rarely happens," (Alexander, p. 70).
We do know that Joshua was Moses' assistant (Ex. 33:11), which quite likely included scribal and editorial duties. He was also the God-appointed successor to Moses (Num. 27:18-21), a prophet whom the Lord spoke to directly (Dt. 34:9, Jos. 1:1-9), and a leader the people obeyed just as they obeyed Moses (Jos. 1:17). We know that Joshua was with Moses when he spoke to God on Mt. Sinai (Ex. 24:13) and in the tent of meeting (Ex. 33:11). We also know that Joshua also contributed to the "Book of the Law of God" (Jos. 24:26). Although not recorded, it is likely that the death of Moses was recorded by Joshua.
- T. Desmond Alexander and David W. Baker, "Authorship of the Pentateuch", by T.D. Alexander, Dictionary of the Old Testament: Pentateuch (IVP, 2003) pp. 61-72
- Bill Arnold and Bryan Beyer, Encountering the Old Testament. (Baker, 1999) pp. 68-9
- Moses: The Author of the Pentateuch? (CRI)
- Mosaic Authorship of the Pentateuch—Tried and True, by Eric Lyons and Zach Smith
- Pentateuch, Catholic Encyclopedia (newadvent.org)