  We present two matched sets of five simulations each, covering five presently
favored simple modifications to the standard cold dark matter (CDM) scenario.
One simulation suite, with a linear box size of 75 Mpc/h, is designed for high
resolution and good statistics on the group/poor cluster scale, and the other,
with a box size of 300 Mpc/h, is designed for good rich cluster statistics. All
runs had 57 million cold particles, and models with massive neutrinos had an
additional 113 million hot particles. We consider separately models with
massive neutrinos, tilt, curvature, and a nonzero cosmological constant in
addition to the standard CDM model. We find that our tilted
Omega+Omega_Lambda=1 (TLCDM) model produces too much small-scale power by a
factor of ~3, and our open Lambda=0 (OCDM) model also exceeds observed
small-scale power by a factor of 2. In addition, we take advantage of the large
dynamic range in detectable halo masses our simulations allow to check the
shape of the Press-Schechter approximation. We find good fits at cluster masses
for delta_c=1.27--1.35 for a Gaussian filter and delta_c=1.57--1.73 for a
tophat filter. However, Press-Schechter overpredicts the number density of
halos compared to the simulations in the high resolution suite by a weakly
cosmology-dependent factor of 1.5--2 at galaxy and group masses, which cannot
be fixed by adjusting delta_c within reasonable bounds. An appendix generalizes
the spherical collapse model to any isotropic cosmology.
