According to Jewish tradition, either Joshua or Ezra inserted, at the end of Deuteronomy, the account of Moses death. The book of Joshua bears his name because it was written by himself. He continued where Moses' portion ended in the Book of the Law:
"Then Joshua wrote these words in the Book of the Law of God. And he took a large stone, and set it up there under the oak that was by the sanctuary of the Lord." (Joshua 24:26)
The book of Judges is generally attributed to Samuel, but it is no real certainty nor of the time it was written.
The authors of the two books of Samuel are not known with certainty. Some attributes them to the prophet Isaiah, while the Pelubert Bible Dictionary refers to different people, including Samuel himself (1Samuel 10:25), Nathan the prophet, Gad or Iddo the seer. Same for the books of Kings. The authors are not known for certainty. Chronicles are traditionally attributed by the Jews to Ezra, and probably correctly so, in spite of modern scholars that believe they were written at a time past Ezra.
Ezra and Nehemiah were written, at least in part, if not mostly, by the same persons. The book of Nehemiah, in particular bears evidence of his own authorship. From the first chapter on he speaks as the narrator of the stories written. Unknown are also the authors of the books of Esther and Job, while the largest portion of Psalms was written by David, and Proverbs by Solomon who also wrote Ecclesiastes and, obviously, the Songs of Solomon.
It goes without saying that the prophets wrote the books bearing their names.
For the New Testament books of the Bible, again, the four gospels were written by the four Evangelists and Acts was written by Luke. Paul wrote most of the so called Epistles, such as Romans, 1st and 2nd Corinthians, Galatians, Ephesians, Philippians, Colossians, 1st and 2nd Thessalonians, 1st and 2nd Timothy, Titus and Philemon. The Apostle Paul likely also wrote the book of Hebrews, although some scholars would contest his authorship of that book. The Apostle Peter wrote his two epistles. The Apostle John, besides one of the gospels, wrote his three epistles plus Revelation and Jude was written by Jude, who was one of Jesus' brothers.