  We report on progress in our study of high temperature QCD with three flavors
of improved staggered quarks. Simulations are being carried out with three
degenerate quarks with masses less than or equal to the strange quark mass,
$m_s$, and with degenerate up and down quarks with masses in the range $0.1 m_s
\leq m_{u,d}\leq 0.6 m_s$, and the strange quark mass fixed near its physical
value. For the quark masses studied to date we find rapid crossovers, which
sharpen as the quark mass is reduced, rather than bona fide phase transitions.
