The most common human mating pattern is monogamy, the next most common polygyny (one husband, two or more wives), then polyandry (one wife, two or more husbands). Tibet had both polyandry and polygyny. I know of no society where group marriages (two or more of each) were common, but examples have existed, such as the Oneida commune in 19th century New York and various smaller groups in the 1960's and thereafter.
Under current U.S. law, although monogamy (including same sex monogamy) is the only form of marriage legally recognized, there do not seem to be any serious legal bars in most states to de facto polygamy. All three forms, although uncommon, exist. It is interesting to speculate on what forms polygamy might take in the future in modern developed societies, given a technological and economic environment different from the environment of past polygamous societies.
On the technological side, two big changes are reliable contraception and paternity testing. The latter solves the most obvious problem with polyandry. Men want to know whether a child is theirs. In the past, the only reliable way of doing so was for the man to have had exclusive sexual access to the child's mother; now all it takes is a reliable lab. So we could have several husbands sharing a wife who bears children by all of them, with each taking a special interest in his own children. The selective use of contraception would even make it possible to decide in advance which men would father children and how many.
One function of marriage is to produce and rear children, another is sexual pleasure. One woman is physically capable of satisfying several men and some women might enjoy doing so. Reliable contraception makes it more practical than in the past to delink sex from marriage, and to a considerable extent it has happened. But keeping sex within a small group has an obvious advantage for reducing the risk of sexually transmitted diseases and may also make better use of the emotional concomitants of intercourse.
Why polyandry instead of monogamy? Modern technology provides ways in which parents can choose to raise the odds of producing a son, and the result in some societies is a substantial m/f imbalance in the population. Polyandry would make it possible in such a society for the excess men to have wives and families.
What about polygyny, the historically more common form of polygamy? It provides a solution to a different problem. Back when legalized abortion and readily available contraception first became hot political issues, a major argument in favor of both was preventing the birth of unwanted children, meaning children born to unmarried mothers. Both legalized abortion and readily available contraception now exist in most developed societies—and have been accompanied by the precise opposite of the predicted effect, a sharp increase in the number of children born to single mothers.
One possible explanation is that, in a world where intercourse was likely to lead to pregnancy, most women were unwilling to sleep with a man unless he was prepared to offer support to any resulting offspring, typically by being engaged or married to her. Breaking that link meant that women who did not want children and did enjoy sex were willing to engage in it without such a guarantee, which sharply worsened the bargaining position of women who wanted sex, children, and support. Some of those who could not find husbands chose to produce children without them. Polygyny would offer some of them the more attractive alternative of being one of two or more wives of men who wanted children and had sufficient resources to support them. And for such men, it provides more sexual variety than monogamy, more emotional security and less medical risk than promiscuity. Also more children.
So far I have been looking at marriage primarily from the standpoint of sex and children. It is also an institution for the production of household services and the sharing of income. In one traditional form of monogamy there was a fairly sharp division of labor, with the wife running the household and the husband working outside it to bring in income. The combination of low rates of infant mortality, meaning that a family that wants to end up with two children need only produce two, and modern technology—washers and dryers, dishwashers, microwave ovens, food bought already prepared—has converted household production from a full time to a part time job. The usual modern response is for both partners to earn income outside of the household, possibly with one of them, usually the wife, taking some years off for child rearing. An alternative, possibly a superior alternative, would be a family of three or more, with one member specializing in running the household and caring for the children.
One additional feature of modern society might affect how practical various forms of marriage would be. We are used to taking it for granted that most income earning is done outside of the household. That has not always been the case in the past—a common pattern has been for family members to be self employed running a shop or a farm. Telecommuting may make something similar again common. The problem of having someone home to keep an eye on the children is substantially reduced if parents who work do their work at home.
All of this is mostly speculation. Over the years I have occasionally encountered people who were part of polygamous families of one sort or another, but have never done much research into how or why they were organized. There is a literature, largely online, on polyamory, but it seems to deal more with structuring the emotional relationships than with organizing production, child rearing, and associated activities.
Comments welcome, especially from those with first hand experience.