I have friends who are in polyamorous relationships, and am involved in one myself based of what most people would classify my relationship as. Polyamory is something I believe is part of human nature. I don't see any reason a person can't love more than one person. People who have multiple children can love all their kids without issues. I don't see any reason a person shouldn't have sex with more than one other person provided all parties are safe, mature, and responsible about it. I think polyamory can be a great thing for the right type of people. Insecure, jealous, and the closed minded should most likely look elsewhere though. Polyamory can raise very well round children (due to being exposed to more views and beliefs), can be more satisfying (due to being able to have different people to meet all of your needs), and is equally beneficial to both genders and all sexualities.
Polygamy on the other hand, is a different entity entirely. It can work, however is asking for problems which others have described. Young men get pushed away and out due to the older males not wanting competition. Young women are "fought over" basically to try and get them first, which leads to younger and younger girls being married. Women tend to get treated poorly while men are the focal point of the culture. I haven't seen many situations where polygamy works.
I don't think that polygamy should be legal. It, more often than not, leads to a society where things happen that most people would view are not good traits. I also don't think that polyamory should be legal based on how our society, laws, and judicial system are setup (although that doesn't mean it should be illegal either).
Polyamory could use some work though to put it in a better spot socially, and it's pretty easy to do too. A lot of polyamorous people I know support the idea of a "primary" relationship and "secondary relationships." What this means is that legally, only the primary spouse is recognized as the spouse. In cases where the secondary relationship has kids, then treat it the same as a current unmarried couple is treated. Both parents (biological ones) get 50/50 split provided there aren't other circumstances like drugs/alcohol/etc. This would require very minimal changes in our laws and judicial system as well. That said, this is really a non-issue in my book. Polyamorous people have learned how to handle the current system, and it generally works. The largest issue is the social issue, which wouldn't be fixed through legislation anyways.
This man wins the thread.
Everyone is looking at this the wrong way around. Polygamy shouldn't be legal, marriage should be illegal. Or just removed from the books.
Why should the government be involved in love? It's ridiculous and anacranistic.
Yup. Remove the word "marriage" from everything related to the government and replace it with "civil union." Then gays can be a part of civil unions just like straight people can without the religious fundies bitching about the "sanctity of marriage" among numerous other benefits from the change.