There can be no doubt that Moses is the author of the Book of Genesis as well as the books of Exodus, Leviticus, Numbers, and Deuteronomy (the Pentateuch). The Jews, which are responsible for preserving the Hebrew Old Testament Scriptures (Romans 3:1-2) ascribe the book of Genesis to Moses.
We also have Jesus’ own testimony that Moses wrote Scripture: “And beginning at Moses and all the Prophets, He expounded to them in all the Scriptures the things concerning Himself” (Luke 24:27). A few verses later Jesus gave the division of the Old Testament and said: “These are the words which I spoke to you while I was still with you, which were written in the Law of Moses and the Prophets and the Psalms concerning Me” (Luke 27:44). Jesus began with Moses because it was Moses who wrote the first five books of the Bible. Further proof is found in John 5:45-47 where Christ said: “Do not think that I shall accuse you to the Father; there is one who accuses you – Moses, in whom you trust. For if you believed Moses, you would believe Me; for he wrote about Me. But if you do not believe his writings, how will you believe My words?”
This does not preclude the fact that Joshua and other prophets added further comments to the law that Moses wrote. Two examples of this are: Genesis 14:14 where the name Dan is used instead of Laish (see Judges 18:29); and Deuteronomy 34:5-12 where we read the account of Moses death.