Generally speaking, the first five books of the Bible,aka the Pentateuch the "Books of Moses": Genesis, Exodus, Leviticus, Numbers, Deuteronomy are considered to be written by Moses. Inspired of God of course, so there are those that would argue thtat God himself is the true author.

There are, however, conflicting theroies:

http://www.religioustolerance.org/chr_tora1.htm

Both Judaism and Christianity assumed that the Pentateuch -- the first five books of the Hebrew Scriptures (Old Testament) were written by Moses, as the Bible itself states. However, in recent centuries, alternative authorship has been proposed. The documentary hypothesis is now accepted by essentially all mainline and liberal theologians.

11th Century CE: Isaac ibn Yashush suggested that the list of the Edomite kings in Genesis 36 was added by an unknown person after Moses died. For this assertion, he became known as "Isaac the Blunderer." 1
15th Century: Bishop Tostatus suggested that certain passages were written by one of the prophets, not by Moses.
16th Century: Andreas van Maes suggested that an editor added additional material to some of Moses' writings.
17th Century: Thomas Hobbes prepared a collection of passages that seemed to negate Moses' authorship.
18th Century: Three investigators (Witter, Astruc and Eichhorn) independently concluded that doublets in the Torah were written by two different authors. A doublet is a story that is described twice, as in: the two creation stories in Genesis
two descriptions of the covenant between God an

Interesting, haven't taken a religion class in a while. Had to look up Elohist.

4:
The Jahwist
The Priestly
The Deuteronomist
The Elohist

or
1:
Moses