In a nutshell: Reincarnation is an important doctrine in Tibetan Buddhism, but is not always embraced by Western Buddhists. Many Buddhists who accept reincarnation believe in the bardo, a spiritual realm where the soul goes between dying and rebirth.
In detail: Lama Surya Das describes the variety of Buddhist views on the afterlife.
Is reincarnation the same as rebirth?
What aspect of us is reincarnated?
If the Self doesn't exist, how can it reincarnate?
Preparing for Reincarnation
Transcript on Tibetan Buddhism from the NPR series "The End of Life: Exploring Death in America."
In a nutshell: Most mainstream Christian denominations reject the concept of reincarnation, citing, among other Bible passages, Hebrews 9:27: "It is appointed for men to die once and after this comes judgment."
In detail: Scripture scholar Ben Witherington III explains that, for Christians, earthly life is a one-time-only affair.
Does the New Testament support reincarnation?
Reading the Elijah and "blind man" scripture passages
Really Born Again: Evangelicals who believe in reincarnation
Debunking "support" for reincarnation in the New Testament
Origen and Beyond: Reincarnation in the history of the early church
In a nutshell: The Roman Catholic Church rejects reincarnation; its catechism states categorically that "Death is the end of man's earthly pilgrimage" and "There is no 'reincarnation' after death" (read citation).
Redemption, Purgatory, and reincarnation
Reincarnation, the early Christians, and body theology
Accessible article on the resurgence of belief in reincarnation, its history, and the Catholic position on it, following the March 1997 Vatican Conference on Reincarnation and the Christian Message.