The Fast of the Tenth of Teves marks the day that Nevuzadran, the Babylonian general, laid siege to Jerusalem prior to the destruction of the first Holy Temple. The siege lasted almost three years until the city walls were breached and the Temple was destroyed.
This was the beginning of a long line of disasters on the Jewish people, including the first exile, and the destruction of the second Temple.
This day is commemorated by refraining from eating or drinking from sunrise to nightfall.
This year, 5770 (2009-2010), Asara B'Teves falls on December 27, 2009.
