There is a fair amount of crass conspiracy theory on social media about this group, In this post, I try to rely on some of the more reliable information.
SABBATEANISM (JEWISH WOMEN'S ARCHIVE)
Sabbateanism—a messianic movement of unprecedented duration and scope—was centred on the charismatic personality of Shabbetai Zevi, a seventeenth-century Jew from the Ottoman port-town of Smyrna who, even after his conversion to Islam in the summer of 1666—a discreditable act which was paradoxically explained in kabbalistic terms as the most challenging part of his mission—was believed by many to be the ultimate redeemer and an incarnate aspect of the kabbalistic godhead. The messianic frenzy he created spread rapidly throughout the Jewish world to become a mass movement, but it subsided gradually following his conversion and evident failure to accomplish his mission by the time of his death in 1676. Nevertheless, as an underground current in kabbalistic Judaism, Sabbateanism persisted until at least the second decade of the nineteenth century. It operated clandestinely, in diverse sectarian groupings, each headed by its own prophet or Messiah who was considered to be a fresh embodiment of the messianic soul that had once inhabited Shabbetai Zevi, and some, like him, were regarded as incarnations of certain aspects of the divine.
The conversion of the purported Jewish messiah Sabbatai Zevi to Islam in 1666 created a crisis among his followers. Many returned to mainstream Judaism; others remained secret Sabbateans. However, a small group in Salonika followed their master into apostasy, converting to Islam in imitation of Sabbatai Zevi. This group, known as the Donme, was very homogenous at the beginning; its members were few in number, knew each other well, and tended to be related to one another. However, shortly after the death of Sabbatai Zevi they split into three factions (a fourth, the Frankists, arose later in Europe), who differed greatly from each other in organization, ritual, and theology.
The Yakubi: Pragmatists and politicians 39
The Karakas: Mystical fraternity 43
Donme in Europe 48
The Kapanci: Merchants and educators 50
EXCERPT: In this, the Karakas serve as a perfect example of a charismatic mode of authority. Osman Baba’s powers are not merely supernatural, but divine; his charisma is such that he is no longer merely a man with supernatural powers but a supernatural being in the form of a man. Their leaders after his death are his descendants, chosen because of their charismatic link to Osman Baba; by sharing in his lineage they share some of his supernatural gifts
See.Osman Baba Karaka for lots more history, including conversions to Catholicism.
SABBATAI ZEVI (Wiki)
Apart from this general Messianic theory, there was another computation, based on an interpreted passage in the Zohar (a famous Jewish mystical text), and particularly popular among the Jews, according to which the year 1648 was to be the year of Israel's redemption by their long-awaited Jewish Messiah.
At age 22 in 1648, Sabbatai started declaring to his followers in Smyrna that he was the true Messianic redeemer. In order to prove this claim he started to pronounce the Tetragrammaton, an act which Judaism emphatically prohibited to all but the Jewish high priest in the Temple in Jerusalem on the Day of Atonement. For scholars acquainted with rabbinical, and kabbalistic literature, the act was highly symbolic. He revealed his Messiahship early on to Isaac Silveyra and Moses Pinheiro, the latter a brother-in-law of the Italian rabbi and kabbalist Joseph Ergas.
However, at this point he was still relatively young to be thought of as an accepted and established rabbinic authority; and his influence in the local community was not widespread. Even though Sabbatai had led the pious life of a mystic in Smyrna for several years, the older and more established rabbinic leadership was still suspicious of his activities. The local college of rabbis, headed by his teacher, Joseph Escapa, kept a watchful eye on him. When his Messianic pretensions became too bold, they put him and his followers under cherem, a type of excommunication in Judaism.
The Dönmeh (Turkish: Dönme) were a group of crypto-Jews in the Ottoman Empire who converted publicly to Islam, but were said to have retained their beliefs. The movement was historically centred in Salonica. The group originated during and soon after the era of Sabbatai Zevi, a 17th-century Jewish kabbalist who claimed to be the Messiah and eventually converted to Islam in order to escape punishment by the Sultan Mehmed IV. After Zevi's conversion, a number of Jews followed him into Islam and became the Dönmeh. Since the 20th century, assimilated Dönmeh might have intermarried with other groups and most have assimilated into Turkish society.
WHY I STUDY SABBATEANISM (Jay Michealson)