Bigamy, Trigamy, Polygamy (Polygyny, Polyandry), Polyamory...

I see absolutely no valid reason for state and federal governments not to legally allow and acknowledge bigamy and polygamy.

This is interesting, though:

There are still states that do not have specific laws defining marriage or outlawing polygamy. In fact, the laws usually defeat themselves. In Utah, law makes polygamy a crime but often polygamy convictions have been thrown out by the courts. The law defines polygamy as someone being married to more than one spouse. But, no one's second marriage is legal if he or she is still married to a first spouse. The law will not reconize a second marriage as valid unless the first marriage is annulled, there's been a divorce, or a death. So, even if someone married a second person by claiming to be single, that marriage would automatically be null and void in the eyes of the law, and being legally married to only one spouse, that person can't be guilty of polygamy. Got it?

In a very real sense, polygamy doesn't legally exist, which is why it's so rarely prosecuted, even in Utah where there are many prosecutors eager to throw polygamists in jail. When it is prosecuted, people are usually charged with something other than actual polygamy.

The answer to legality/illegality seems to be that in many states, polygamy is not illegal, at least not a crime. This "legality" is often based on case law where statutes exist making polygamy a crime. The important question, however, is whether polygamous marriage is accepted legally as being equal to traditional marriage.
Constitutional guarantees of personal freedoms, religious freedoms, equal treatment under the law, and the legal contradictions and impossibilities in anti-polygamy laws, combine to make it nearly impossible to actually "outlaw" polygamy in any meaningful way.