From the current (2017) LDS Handbook 2: Administering the Church:
Surgical Sterilization (Including Vasectomy)
The Church strongly discourages surgical sterilization as an elective form of birth control. Surgical sterilization should be considered only if (1) medical conditions seriously jeopardize life or health or (2) birth defects or serious trauma have rendered a person mentally incompetent and not responsible for his or her actions. Such conditions must be determined by competent medical judgment and in accordance with law. Even then, the persons responsible for this decision should consult with each other and with their bishop and should receive divine confirmation of their decision through prayer.
Other relevant sections from Handbook 2:
The Church strongly discourages artificial insemination using semen from anyone but the husband. However, this is a personal matter that ultimately must be left to the judgment of the husband and wife. Responsibility for the decision rests solely upon them.
Artificial insemination of single sisters is not approved. Single sisters who deliberately refuse to follow the counsel of Church leaders in this matter are subject to Church discipline.
Chastity and Fidelity
The Lord’s law of chastity is abstinence from sexual relations outside of lawful marriage and fidelity within marriage. Sexual relations are proper only between a man and a woman who are legally and lawfully wedded as husband and wife. Adultery, fornication, homosexual or lesbian relations, and every other unholy, unnatural, or impure practice are sinful. Members who violate the Lord’s law of chastity or who influence others to do so are subject to Church discipline.
Homosexual Behavior and Same-Gender Attraction
Homosexual behavior violates the commandments of God, is contrary to the purposes of human sexuality, and deprives people of the blessings that can be found in family life and in the saving ordinances of the gospel. Those who persist in such behavior or who influence others to do so are subject to Church discipline. Homosexual behavior can be forgiven through sincere repentance.
If members engage in homosexual behavior, Church leaders should help them have a clear understanding of faith in Jesus Christ, the process of repentance, and the purpose of life on earth.
While opposing homosexual behavior, the Church reaches out with understanding and respect to individuals who are attracted to those of the same gender.
If members feel same-gender attraction but do not engage in any homosexual behavior, leaders should support and encourage them in their resolve to live the law of chastity and to control unrighteous thoughts. These members may receive Church callings. If they are worthy and qualified in every other way, they may also hold temple recommends and receive temple ordinances.
In Vitro Fertilization
The Church strongly discourages in vitro fertilization using semen from anyone but the husband or an egg from anyone but the wife. However, this is a personal matter that ultimately must be left to the judgment of the husband and wife. Responsibility for the decision rests solely upon them.
The Church strongly discourages the donation of sperm.
The Church strongly discourages surrogate motherhood. However, this is a personal matter that ultimately must be left to the judgment of the husband and wife. Responsibility for the decision rests solely upon them.