Did Moses write the first five books of the Bible?

yes Later rabbis (and the Talmudic rabbis as well - see tractate Bava Basra 15a) and Christian scholars noticed some difficulties with the idea of Mosaic authorship of the entire Torah, notably the fact that the book of Deuteronomy describes Moses' death; later versions of the tradition therefore held that some portions of the Torah were added by others - the death of Moses in particular was ascribed to Joshua. The Talmud explains this by saying that Moses wrote it tearfully, in anticipation of his death; another tradition is that Joshua added these Words after Moses died (the next book is the Book of Joshua which, according to Jewish tradition, was written by Joshua himself), and that the final verses of the book of Deuteronomy read like an epitaph to Moses.

"Mosaic authorship was accepted with very little discussion by both Jews and Christians until the 17th century, when the rise of secular scholarship and the associated willingness to subject even The Bible to the test of reason led to its rejection by mainstream biblical scholars. The majority of modern scholars believe that the Torah is the product of many hands, stretching over many centuries, reaching its final form only around the 6th and 5th centuries BCE."