I understand that polygamy was commonly taught and practiced in the early days of the LDS church by Joseph Smith and others. I'm wondering just how many wives he had and who they were. Also, were any of these women polyandrous (having more than one husband) or was only polygamy (more than one wife) allowed?
35 according to historian Todd Compton.
43 according to author George D. Smith.
47 according to author Fawn Brodie.
Some of his wives seem to have had additional husbands.
Polygamy is one of the more difficult subjects to study of early LDS history, because the environment in which it was practiced necessitated secrecy as a result of the hostility toward the Church and it's usefulness as a polemical tool for the enemies of the Church. However, credible research is continually trickling out, and we now have some great theories to explain it. Unfortunately there is enough contradictory evidence that we may never know with 100% certainty what actually happened.
Rather than attempting a full explanation, I'll provide some brief thoughts and references to external sources.
I believe the number of wives is around 27 to 29, with 3-5 being wives he spent any amount of time with. There are a lot of different opinions out there, so you'll likely get a different answer depending on who you talk to. FAIRMormon has some excellent documentation on the subject.
Dr. Greg Smith of the Foundation for Apologetic Information and Research published a fantastic paper in 2005 called, "Polygamy, Prophets, and Prevarication". While not specifically commenting on the number of wives Joseph had, he discusses a lot of the historical and contemporary issues surrounding polygamy and why there are so many questions surrounding this.
In 2012, Brian Hales gave a great speech where he also addresses how many wives he thinks Joseph had, including how many were polyandrous marriages. He also discusses some of the background and theology surrounding the polyandrous marriages, providing some good primary research that he has written about in his multi-volume books.
Richard Lyman Bushman spends a lot of time discussing Joseph's wives and investigating the issues surrounding them in his excellent biography of Joseph called Rough Stone Rolling.