Religious teachings

Wikipedia, in its roundup of reincarnation says that Ian Stevenson reported that belief in reincarnation is held (with variations in details) by adherents of almost all major religions except Christianity and Islam. In addition, between 20 and 30 percent of persons in western countries who may be nominal Christians also believe in reincarnation

Buddhism

Reincarnation is a fundamental teaching of Buddhism.  There is a lot of dispute about the detail.
Reincarnated teachers are recognized all over the world.  As these have been born outside Nepal, problems with integrating the past and present have arisen.

Christianity

Reincarnation was allegedly excised from the bible in the time of Constantine. He didn't do a very good job, however, as there are a lot of references to rebirth and to past lives. John the Baptist was said to be the reincarnation of Elijah, and so on through some very dreary yes he was no he wasn't sorts of discussions. There are lots of web pages which discuss what may or may not be known about reincarnation in the Christian tradition.

Hinduism 

Reincarnation is an intrinsic part of Hinduism, in which birth, death and rebirth are an important tenet.  

Islam 

Although mainstream Islam rejects the doctrine of reincarnation, there are a number of mystical and sufi groups which include it as a part of their beliefs.

Judaism

While reincarnation is not an essential tenet of traditional Judaism, it forms an essential part of Kabbalah, the mystical school of teaching.

Wicca

Many Wiccans believe in reincarnation, but it isn't an essential tenet of belief.  It is hard to say what those might be, any way, as there is no central authority.

Ourside the main religions in the west, there is widespread belief in reincarnation. Hinduism, Buddhism and many others believe in the reincarnation of the soul. Some provide information about the rules and regulations, the role of karma, etc.

Modern new age thinking incorporates elements of everything in a soup of belief that can be wide enough to encompass parallel lives, soul groups, pre-birth contracts between individuals or groups and a whole lot more.