  We investigate the gravitational collapse of a spherically symmetric, perfect
fluid with equation of state P = (Gamma -1)rho. We restrict attention to the
ultrarelativistic (``kinetic-energy-dominated'', ``scale-free'') limit where
black hole formation is anticipated to turn on at infinitesimal black hole mass
(Type II behavior). Critical solutions (those which sit at the threshold of
black hole formation in parametrized families of collapse) are found by solving
the system of ODEs which result from a self-similar ansatz, and by solving the
full Einstein/fluid PDEs in spherical symmetry. These latter PDE solutions
(``simulations'') extend the pioneering work of Evans and Coleman (Gamma = 4/3)
and verify that the continuously self-similar solutions previously found by
Maison and Hara et al for $1.05 < Gamma < 1.89 are (locally) unique critical
solutions. In addition, we find strong evidence that globally regular critical
solutions do exist for 1.89 < Gamma <= 2$, that the sonic point for Gamma_dn =
1.8896244 is a degenerate node, and that the sonic points for Gamma >Gamma_dn
are nodal points rather than focal points as previously reported. We also find
a critical solution for Gamma = 2, and present evidence that it is continuously
self-similar and Type II. Mass-scaling exponents for all of the critical
solutions are calculated by evolving near-critical initial data, with results
which confirm and extend previous calculations based on linear perturbation
theory. Finally, we comment on critical solutions generated with an ideal-gas
equation of state.

General Relativity and Quantum Cosmology