Karaites
Karaites are members of a Jewish sect that separated from rabbinic Judaism around 800 CE. Their faith is often referred to as Karaism, or Karaite Judaism.
Karaites are often confused with the Ananites, founded by Anan ben David, however it can be seen from the writings of Jacob Al-Kirkisani, and other Karaite Sages, that Anan ben David did not found Karaism. Karaism developed under the leadership of Benyamin al-Nahawendi, Daniel al-Qumisi, and Jacob Al-Kirkisani (See Leon Nemoy, Karaite Anthology, Introduction).
Karaism relies solely on the Tanakh (Hebrew Bible) as the basis for religious law. They reject the authority of the Jewish oral law, and as such in most cases they do not rely on the Mishna and Talmud, or any of the rabbinic works that stem from the Talmud.
Karaite calendar
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.
Karaite Writings
Like Rabbinic Judaism, Karaism has produced a vast library of writings and polemics. While tradionally known for its anti-Rabbinic literature, Karaism has produced quality peshat commentaries (which are referenced in the commentaries of Rabbi Abraham ibn Ezra), and is known for the first Counter-Missionary materials, such as 'Faith Strengthened' by Karaite Isaac Troki which is the basis of all modern Counter-Missionary material. Scholary studies of Karaite infancy, as Leon Nemoy notes, is in its infancy.
In the State of Israel
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 Separatism in 19th Century Russia).