There are many others that are required by Rabbinic law, but not torah law. In fact we just passed one, Tisha B'av, the ninth of Av, when many terrible things happened to the Jewish people, individually and as a nation. Two major ones being this was the day that the Temple was destroyed, both Solomons' and Herods. Also the day that the Jews were ordered out of Spain ( one day before Christopher Columbus sailed from Spain, he is believed by some to have been a Cyrpto Jew). So there is fasting and mourning because the temple was destroyed.
But the required fast day, a day to afflict your soul, and to turn from fleshly things and concentrate wholely on spirtual is yom kippur, the day of atonement. This day is coming up soon. Beginning on the 8th of October at sundown.