  The degrees of freedom (DoF) region of the fast-fading MIMO (multiple-input
multiple-output) Gaussian broadcast channel (BC) is studied when there is
delayed channel state information at the transmitter (CSIT). In this setting,
the channel matrices are assumed to vary independently across time and the
transmitter is assumed to know the channel matrices with some arbitrary finite
delay. An outer-bound to the DoF region of the general $K$-user MIMO BC (with
an arbitrary number of antennas at each terminal) is derived. This outer-bound
is then shown to be tight for two classes of MIMO BCs, namely, (a) the two-user
MIMO BC with arbitrary number of antennas at all terminals, and (b) for certain
three-user MIMO BCs where all three receivers have an equal number of antennas
and the transmitter has no more than twice the number of antennas present at
each receivers. The achievability results are obtained by developing an
interference alignment scheme that optimally accounts for multiple, and
possibly distinct, number of antennas at the receivers.
