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Tomchei Tmimim (Hebrew: תומכי תמימים, "supporters of the complete-wholesome ones") is the central Yeshiva (Talmudical academy) of the Chabad-Lubavitch Hasidic movement. Founded in 1897 in the town of Lubavitch by Rabbi Sholom Dovber Schneersohn, it is now an international network of institutions of advanced Torah study, the United Lubavitcher Yeshivoth.
See also: Yeshiva § Hasidic yeshivas
As above, Tomechei Tmimim was founded in 1897 in Lubavitch, by Rabbi Sholom DovBer Schneersohn for the study of Hasidic philosophy according to the Chabad tradition, in parallel with the traditional Yeshiva curriculum. Here, Rabbi Schneersohn authored Kuntres Eitz HaChayim, guidelines and standards for a student's learning goals and schedule, personal conduct, prayer, and appearance. Correspondingly, he called the students of this yeshiva "tmimim" (sing. "tomim" תמים = pure, perfect).
When Rabbi Yosef Yitzchok Schneersohn left the Soviet Union in 1927, the yeshiva reestablished itself in Warsaw and later in Otwock, Poland. In the course of World War II, the yeshiva escaped to Shanghai, China, along with some other yeshivot like Mir.
Once the Rebbe was safely evacuated to New York, the Yeshiva was reestablished in New York City, where it remains to this day. Within 24 hours the Rebbe had opened a yeshiva branch. Starting with 10 students, the Yeshiva quickly grew to the extent that it expanded to other nearby locations, giving rise to the group known as "United Lubavitcher Yeshivas"; see next section. Rabbi Yisroel Yitzchok Piekarski was the Rosh Yeshiva for 42 years, from 1951 until 1993.
The central Yeshiva is housed today at the Chabad Lubavitch World Headquarters, at 770 Eastern Parkway in Brooklyn, NY, with approximately six hundred students. Similarly named yeshivas, many of which are nevertheless formally independent, are to be found in major cities in the United States, Canada, Europe, South America, South Africa, Australia, and the former Soviet Union, and Israel. (Not all carry the name "Tomchei Tmimim" however.)
Many of the branches also perform the functions necessary to grant Semicha, rabbinical ordination, to their students;   true since the 1950's for the majority of students. Some affiliated institutions  (often essentially “post-graduate”) specifically focus on Rabbinical training, with the Semicha correspondingly requiring further depth and breadth.
A significant number of graduates of Tomchei Tmimim continue working within Chabad as religious functionaries, whether as shluchim in Chabad Houses or as teachers in schools. Graduates - usually of the latter institutions mentioned - also often work as "community Rabbis" more broadly.