  We present an update of our study of high temperature QCD with three flavors
of quarks, using a Symanzik improved gauge action and the Asqtad staggered
quark action. Simulations are being carried out on lattices with Nt=4, 6 and 8
for the case of three degenerate quarks with masses less than or equal to the
strange quark mass, $m_s$, and on lattices with Nt=6 and 8 for degenerate up
and down quarks with masses in the range 0.2 m_s \leq m_{u,d} \leq 0.6 m_s, and
the strange quark fixed near its physical value. We also report on first
computations of quark number susceptibilities with the Asqtad action. These
susceptibilities are of interest because they can be related to event-by-event
fluctuations in heavy ion collision experiments. Use of the improved quark
action leads to a substantial reduction in lattice artifacts. This can be seen
already for free fermions and carries over into our results for QCD.
