Karaism relies solely on the Tanakh (Hebrew Bible) as the basis for religious law. By implication, they do not rely on the Talmud (which would inlude the Mishnah), or any of the rabbinic works that stem from the Talmud.
Karaites rely on observations of the Moon to begin their months, and on observations of barley (called the Aviv) to begin their years, as deduced from statements in the Torah (Aviv[?] is both marker for the first month of the Biblical Hebrew calendar, and the next-to-last stage in the growth of barley, which it was in during the plague of hail shortly before the first Passover). Before quick worldwide communication was available, Karaites in the Diaspora used a variety of methods to determine the calendar, including observation and calculation.
The state of Israel formally recognizes all Karaites as Jews, while not all Karaites, particularly those from Eastern Europe, formally recognize themselves as Jews. Rather, they consider themselves ethnically Turkic (See Philip Miller, Karaite Seperatism in 19th Century Russia).
See also: Judaism