An Official Declaration found at the end of the Doctrine and Covenants issued by the Mormon prophet Wilford Woodfuff in the 1890s ended the practice of plural marriages in the Mormon church. Prior to Official Declaration 2, polygamy had been openly practiced by the members since the time of Brigham Young. Genealogical records found on the official website for the Church of Jesus Christ of Latter Day Saints shows Joseph Smith was married more than one wife during his life time.
Today, the Mormon church excommunicates any member found to have more than one wife and many Mormons believe it to no longer be a part of Mormon doctrine. A Reading of the Doctrine and Covenants shows that the doctrinal aspects of polygamy still live in the scripture of the religion.
While the Book of Mormon forbids the practice of Plural Marriage, Doctrine and Covenants Section 132 commands Mormons to practice polygamy. The scriptural verso also states that plural marriage is needed to enter the new and everlasting covenant. Joseph Smith tells his wife, Emma, that she can either except the new and everlasting covenant or be destroyed. Polygamy would continue to be practiced by the Mormon hierarchy in secret until Brigham Young declared to the world that the Mormons were openly practicing polygamy in 1852.
Increasing pressure from the U.S. Government to stop the practice of plural marriage led up to the official declaration of the 1890s. Polygamy had kept Utah, which had long since passed the requirements for statehood from becoming an official part of the United States.
Mormons can only have one living wife at a time today, but a form of spiritual polygamy still exists. A man who holds the priesthood can be sealed to more than one woman in the temple, provided that his first wife is dead. It is also possible if the man got a a divorce happened that had not been followed up by following the necessary paperwork with the Mormon church itself that a divorced wife may still be bound to her husband in the afterlife in the eyes of the church.
The Church of Jesus Christ of Latter Day Saints is unlikely to change its position on this form of polygamy as it conforms to the laws of the land and polygamy being practiced in the afterlife only matters if the Mormons are right.
Sources:
Doctrine and Covenants. Section 132.
Doctrine and Covenants, Official Declaration 2
Journal of Discourses.
Book of Mormon, Mosiah