Did Moses really write Genesis? Is he the author of the "Books of Moses," the Old Testament Law?
Jewish tradition holds that Moses either wrote or was responsible for the whole Pentateuch (the five rolls, Genesis-Deuteronomy), which is the Torah (O.T. Law). Accordingly, in some translations—for example, the German, Swedish, and Bulgarian—Genesis is called "The First Book of Moses," while other translations, like the French, Portuguese, and Spanish, use an equivalent of "Genesis."
That Moses wrote the first five books of the Bible is the traditional view, though with fewer supporters nowadays than one or two hundred years ago. We know Moses did not write the end of the "Fifth Book of Moses" (Deuteronomy 34 recounts Moses' own death), and it is unlikely that he wrote the comment on his character in Numbers 12:3. So even if Moses was substantially responsible for writing the Torah, it is clear that he was not the final editor.
This is easily demonstrable in the case of Genesis, whose final form could not have been written any earlier than about 950 BC. We know this because of several otherwise anachronistic references, for example the mention of kings in Israel (36:31). Whereas the Scriptures do not portray Moses as writing Genesis, they certainly do portray him maintaining a book of the law in which he recorded the core of the teachings of the Torah. Does it really matter whether or not Moses wrote Genesis? Not to me, though some readers may feel differently.
If you are new to biblical criticism, at first the notion that books of the Bible were sometimes composed gradually may be disturbing. I would answer that works of art are often carved, painted or written in stages. Great edifices go up in stages, and though the credit may go to one man, or to the chief architect, the endeavor is often a team effort. Many books of the Bible certainly do have different layers, or multiple authors, and this is easily demonstrated. The problem is that the word of God is reduced to the word of man when the conclusion is drawn that human influence precludes divine influence.
The Books of Moses, according to the preponderance of biblical critics, show "mosaic" authorship. Not "Mosaic" in that they pertain to Moses (one definition of "Mosaic"), but "mosaic" because it is believed they were written and rewritten by different authors with conflicting theologies. The Pentateuch, according to these theorists, is a patchwork, in effect.