In a nutshell: The LDS Church teaches that there is only one physical death for any one person. Within a lifetime, each person experiences spiritual "births," but these "do not in any way constitute any type of reincarnation," according to the Encyclopedia of Mormonism. After death, existence in a spirit world precedes the resurrection. Eventually those who lived the gospel of Jesus will have a glorified celestial body.
Reincarnation and the Book of Mormon
LDS Theology Rejects Reincarnation
In a nutshell: Eastern Orthodoxy teaches that "The moral progress of the soul, either for better or for worse, ends at the very moment of the separation of the body and soul; at that very moment the definite destiny of the soul in the everlasting life is decided. ...There is ...no reincarnation" (read citation).
In a nutshell: The concept of reincarnation is thought to have originated within Hinduism, the world's oldest major religion. Hindus believe they will experience multiple rebirths based on their actions in this and previous lives. This doctrine of cause and effect over many lives is called karma.
In detail: The remarkable story of Shanti Devi, a four-year-old girl whose distinct past life memories were researched by several investigators, illustrates what Hindus believe about reincarnation.
When the cycle finishes, is there a Hindu heaven?
Hindus skeptical about reincarnation
Will we reincarnate as members of other religions?
Deepak Chopra responds
Karma as DNA
Are inherited (genetic) illnesses the result of bad past life karma?
In a nutshell: Muslims reject the concept of reincarnation, believing that the soul will be judged after death.
In detail: Kenneth Briggs explains what Muslims believe about the Day of Judgement.
Can any Qur'an passages be interpreted to support reincarnation?
Does God give each person enough time for tasks to be completed?
The Qur'an and Reincarnation
Argues that the Qur'an unequivocally rejects the idea of reincarnation.
In a nutshell: Because Judaism is primarily focused on life here and now, it puts less emphasis on afterlife teachings. Some Jewish mystical traditions accept the concept of gilgul, an ongoing cycle in which the soul is reborn. However, most mainstream Jews reject the concept of reincarnation.
In detail: Rabbi David Wolpe explains Jewish views of the afterlife.
Gilgul: Recycling the soul
Resurrection vs. reincarnation
Olam Ha-Ba: The Jewish Afterlife
Did Biblical Characters Reincarnate?
One rabbi's list of those reincarnated in Jewish history.