Polygamy is a marital practice in which a person has more than one spouse simultaneously. The term polyamory has been used in recent years to refer to being in romantic or sexual relationships with multiple partners at once; it need not involve marriage. Polygyny refers specifically to one man having multiple wives, and polyandry to one woman having mutiple husbands. Historically, both practices have been found in many cultures, but polygyny is far more common than polyandry.
Polygynous societies are about four times more numerous than monogamous ones. In 1994, Theodore C. Bergstrom noted in his paper "On the Economics of Polygyny"  (U. Mich. Center for Research on Economic and Social Theory, Working Paper Series 94-11) that "Although overt polygamy is rare in our own society, it is a very common mode of family organization around the world. Of 1170 societies recorded in Murdock's Ethnographic Atlas, polygyny (some men having more than one wife) is prevalent in 850."
Polygamy is currently prohibited by almost all Jewish and Christian groups. It was permitted in early Judaism, as can be seen from the ancient Patriarchs and the Jewish Kings, although only a minority of Jews practiced it. It was forbidden by Roman law, and by the time of the rise of Christianity, nearly all Jews had abandoned the practice. It was never permitted in early Christianity.
There are few pronouncements of the early Christian church that explicitly prohibit polygamy, since it was almost unheard of in Graeco-Roman society. Early Christians desired to condemn polygamy, because it conflicted with the prevailing mores of the Graeco-Roman society in which they lived; yet at the same time they had to explain the clear permission given for it in the Old Testament. Saint Augustine demonstrates this conflict in his consideration of the polygamy practiced in the time of the Old Testament patriarchs when he writes "The Good of Marriage" (chapter 15, paragraph 17) that though it "was lawful among the ancient fathers: whether it be lawful now also, I would not hastily pronounce. For there is not now necessity of begetting children, as there then was, when, even when wives bare children, it was allowed, in order to a more numerous posterity, to marry other wives in addition, which now is certainly not lawful." He declines to judge the patriarchs, but he certainly makes the current illegality clear. In another place, he wrote, "Now indeed in our time, and in keeping with Roman custom, it is no longer allowed to take another wife, so as to have more than one wife living [emphasis added]."
In the 5th century both Pope Innocent I and Pope Hilarus issued explicit decrees against polygamy. As the Church spread into more cultures that had previously allowed polygamy, the magisterium continued to condemn the practice, though they also continued to struggle with the theology of the prohibition. Martin Luther wrote in De Wette, II, p. 459, "I confess that I cannot forbid a person to marry several wives, for it does not contradict the Scripture. If a man wishes to marry more than one wife, he should be asked whether he is satisfied in his conscience that he may do so in accordance with the word of God. In such a case, the civil authority has nothing to do in such a matter."
The Catholic Church clearly condemns polygamy today; the Catechism of the Catholic Church lists it in paragraph 2387 under the head "Other offenses against the dignity of marriage" and states that it "is not in accord with the moral law." (((I'm sure all Protestant churches do, too -- can someone specify?))) Splinter groups of the Church of Jesus Christ of Latter-Day Saints, some other small non-LDS Christian groups in the United States, and some African Christians also practice polygamy. Yemenite Jews continued to practice polygamy up until the 1950s.
Polygamous marriages are not recognized by secular law in most "Western" countries with large Jewish and Christian populations. However, few such countries have any laws against living a polygamous lifestyle: they simply refuse to give it any official recognition. Parts of the United States, however, criminalise even the polygamous lifestyle, which is unusual; these laws originated as anti-Mormon legislation, although they are rarely enforced.
Islam, unlike Judaism and Christianity, never prohibited polygamy. It provides legal requirements and restraints that amount to the discouragement of the practice. The verse of the Qur'an that allows polygamy can be translated as "If you fear that you shall not be able to deal justly with the orphans, marry women of your choice, two, or three, or four; but if you fear that you shall not be able to deal justly, then only one...." (Qur'an 4:3) Another verse in the same chapter discourages polygamy, stating: "You are never able to be fair and just between women even if it is your ardent desire...." (Qur'an 4:129) The application of these principles stated in most discussions of Islamic polygamy is that a man is required to make equal provision ('deal justly') for each wife and to treat their children equally as well. Several Islamic countries, however, prohibit polygamy anyway (e.g. Libya, Tunisia), and many moderate or liberal Muslims are opposed to the practice. Islamic scholars are divided as to whether an Islamic ruler can prohibit polygamy.
Polygamy was practiced in the United States by the early Church of Jesus Christ of Latter-day Saints, who refered to it as plural marriage. It led to persecution of the LDS, and the prohibition of the practice after August 29, 1852, in Utah fanned public hostility against the Church. Although Latter-day Saints believed that their religiously-based practice of plural marriage was protected by the United States Constitution, opponents used it to delay Utah statehood until 1896. Increasingly harsh antipolygamy legislation stripped Latter-day Saints of their rights as citizens, disincorporated the Church, and permitted the seizure of Church property before the Church ordered the discontinuance of the practice in 1890.
National attention in the United States again focused on potential polygamy among the LDS in the early 1900s during the House hearings on Representative-elect B. H. Roberts and Senate hearings on Senator-elect Reed Smoot (the Smoot Hearings), which caused LDS President Joseph F. Smith to issue his "Second Manifesto" against polygamy in 1904. Since that time, it has been Church policy to excommunicate any member either practicing or openly advocating the practice of polygamy.