In Hebrew it is called gilgul. Maimonides discusses Daniel 12:13
“now go your way to the end and rest, and you shall arise to your destiny at the end of days.”
There is also something in Judaism called “yibum”. If a married man dies childless, tradition holds that his spouse should marry the man’s brother and their first-born should receive the dead father’s name.
Rabbi Haim Vital compiled a list of those reincarnated in Jewish history. There is a cycle of reincarnations beginning with Dinah and Shechem. Dina, the daughter of Jacob was raped by Shechem. Shechem did not take responsibility for his actions and blamed them on his upbringing and the fact that Dinah was a noble women. So the role were reversed when Shechem was reincarnated as Zimri, an Israelite general, and Dina as Cuzbi, a Midianite women. Zimri was found consorting with Cuzbi and both were killed by the zealot, Pinhas. Thus when Shechem/Zimri was a noble man and of good birth, he could no longer blame outside sources for his own faults and was punished accordingly. The story continues when Pinhas was reincarnated as Rabbi Akiva and Cuzbi, as the wife of the Roman general Turnus Rufus. She converted to Judaism and helped establish the yeshiva of Rabbi Akiva. By promoting Jewish learning in her next life, she atoned for his sins with Zimri. Thus perhaps ending that cycle of gilgul.
A midrash says that every Jew to ever live, and who ever will live, stood at Mount Sinai when the Jews received the Covenant from God. Reincarnation may help explain how this midrash could be true.
As I said, this is not part of normative judaism and reasonable jews will differ. The Chassidim probably have a stronger belief in reincarnation than other jews. I got the above information from the Jewish Virtual Library.
