Who is thought to be the author of the Book of Job?

The authorship of the book of Job is probably the most uncertain of any in the Bible. Not only can we not determine WHO wrote it, it's not even certain WHEN.
Most commentators I know of believe the events to have taken place around the time of Abraham, maybe 2000 BC. The lifestyle wherein the father is the priest of the family, and wealth is measured in livestock, are such indications. Additionally, there is no mention of Abraham, or of Job's relationship to him, if any; nor of Moses, the law, the Temple, or the kings.
The book in the form we have today may have been written as late as 500 BC. Some people see a post-exilic sensitivity in it--I'm not that astute a scholar. So it could have been written by almost anyone from Abraham's time on, maybe by some second-temple-era scribe, working from ancient sources.
So far as I know, the Jews never questioned its authenticity as an inspired work, so it would appear that whover first presented it, did so with satisfactory bona fides, even if that information is now lost to us.

--Thanks for making sure we're clear on that Glenn, I absolutley agree. I take the book of Job to be describing real events--both in heaven and on earth--giving us real, direct insight into the activities of God and Satan; a parable or allegory would not serve the purpose. The uncertainty about who wrote down the preserved account, and when, need not call that point into question.

Various authors have been suggested throughout history, settling with the main possibilities being Moses, Solomon, Elihu or even Job himself. Unger assigns Moses as the author, and there are quite a number of people who hold to this. Nevertheless, it cannot be absolutely determined.

Author as already been stated is unknown, but the message is timeless, that despite our lack of knowledge we can and we must have faith.

Job
Writer: Moses
Place Written: Wilderness
Writing Completed: c. 1473 B.C.E.
Time Covered: Over 140 years between 1657 and 1473 B.C.E.
5 Why do we say Moses was the writer? This is according to the oldest tradition, among both Jewish and early Christian scholars. The vigorous authentic style of Hebrew poetry used in the book of Job makes it evident that it was an original composition in Hebrew, the language of Moses. It could not have been a translation from another language such as Arabic. Also, the portions in prose bear stronger resemblance to the Pentateuch than to any other writings in the Bible. The writer must have been an Israelite, as Moses was, because the Jews “were entrusted with the sacred pronouncements of God.” (Rom. 3:1, 2) After he had reached maturity, Moses spent 40 years in Midian, not far from Uz, where he could obtain the detailed information recorded in Job. Later, when he passed near Job’s homeland during Israel’s 40-year wilderness journey, Moses could learn of and record the concluding details in the book.

The Holy Spirit.