Reincarnation, gilgul in Hebrew, is not explicitly mentioned in the Torah, though some interpret traditional Jewish practices to refer to reincarnation. Maimonides has written about the allusion of biblical verses to its occurrence. For example, Daniel 12:13 states, "now go your way to the end and rest, and you shall arise to your destiny at the end of days." A second example can be found in the concept of yibum, when a married man dies childless, tradition holds that his spouse should marry the mans brother and their first-born should receive the dead fathers name.
The purpose of reincarnation is seen as a chance for a soul to achieve a goal not achieved in a previous life and as a chance to reward man for fulfilling the desires of his Creator. Reincarnation has also been viewed as punishment for a sinners previous deeds. For example, a rich man who abused his power may come back as poor.
Rabbi Haim Vital, a student of the ArI, has 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.
Sources: Dubov, Rabbi Nissan Dovid. An
Overview of Techiyas Hameisim Based on the Teachings of Chabad Chassidism,