Gilgulim (Reincarnation)While some rationalist philosophers as Rav Saadia Gaon doubted the possibility of gilgulim, those Jewish philosophers more prone to Jewish mysticism, not only accepted this possibility but advocated for it as did Rabbi Chaim Vital the tamid mevuhakh (disciple) of HaAri HaKodesh in Sefer HaHezyanot (Book of Visions). In Lurianic Kabbalah souls under Tikkunim to make a repair for lacunai in previous generaitons. Unlike the many souls which are subject to the general laws of transmigration, high-ranking souls of a Tzadik Gemur were thought to remain concealed among the "supernal lights" until their time arrived and not to enter the cycle of reincarnation at all but transmorgrafied on departure, to reside under the throne of Hashem. Thus there are different types of souls (5 words for soul in Hebrew) but the term "original souls" (neshamot mekoriyyot) came to take on special significance in Kabbalah although it is thought that Adam Rishon contained all the souls of human history etc Besides Lurianic Kabbalah, The early 5th C midrash also considers the existence of reincarnation for instance in the case of Yonah which the GRA in the 18th century notes (see power point by DBL on Sefer Yonah and Ninevah). Elaborate logical systems that are rationally coherent developed in Jewish thought linking "soul sparks" based on some common characteristic in the feature of two persons inner neshamah. For example Pinchas and Eliyahu HaNavi since they possesed the characteristic of Zealissness (ziziyroot) are said to be gilgulim. Later after Biblical History arguments developed for instance that since Rabbi Akiva was martryred at the age of 40 (departed at 40 kidush Hashem) and the Ramhal lived to the age of 40, ergo the Ramhal is a gilgul of Rabbi Akiva. Also logical links for "soul spark transference" are made by uncanny coincidences in two individuals lives that attain to a connection behind "the veil" that links their soul spark evolution. Sometimes "families of soul sparks" have a mission to make Tikkunim in later generations based on relationships of the soul sparks had in previous generations.
The Light Beyond by 
Moses Maimonides' Treatise on Resurrection by 
Journey to Heaven by