There are some books of the Bible whose authors cannot be known completely.
The first five books of the Bible, - Genesis, Exodus, Leviticus, Numbers and Deuteronomy, are referred to in other parts of the Bible and by Jesus himself, as the books of Moses.
It is accepted that Joshua (or a scribe on his behalf) wrote the book of Joshua. Then various people wrote Judges,1 Samuel-2 Chronicles which appears to have been maintained as an official record. The authors of Ruth, Esther and Job are unknown. The Psalms were mostly written by king David, Solomon wrote Proverbs Ecclesiastes and Song of Solomon. The rest of the prophetical books were written by the people who gave their names to the books e.g Daniel, Ezekiel.
In the New Testament, Matthew, Mark, Luke and John were written by the men of those names; there are some variations on who people think wrote the Acts, but the most probably author seems to be Luke.
Paul wrote the following letters/epistles:
Romans, 1 & 2 Corinthians, Galatians, right through to include Philemon.
The author of the Letter to the Hebrews is not identified, but thought to also be from Paul. James, 1 & 2 Peter, 1,2,& 3 John and Jude were written by people of those names. The Revelation was written by John.



Comments
For anyone that believes the Bible, Jesus saying that Moses wrote them would be proof enough... Otherwise, aren't you inferring that Jesus was/is a liar?
I say that "Thomas Jefferson was the principle author of the US declaration of independence". You are correct that this is not proof. However, it is accepted as true. So even in Jesus' time, the Jewish people accepted this as truth as do the parts of the old testament referred to by the answerer. Now, I agree with Cly. If this is not true, then what else should a person not believe about Jesus' words? By his claims, we must either deduce that he lies or he has the authority to ascribe this to Moses.
2 Timothy 3:16 All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness
this is also the reason why John was able to write the book of Revelations
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