  The distributions of peculiar velocities of rich clusters and of groups of
galaxies are investigated for different cosmological models and are compared
with observations. Four cosmological models are studied: standard ($\Omega=1$)
CDM, low-density ($\Omega=0.3$) CDM, HDM ($\Omega=1$), and PBI ($\Omega=0.3$),
all COBE normalized. We find that rich clusters of galaxies exhibit a
Maxwellian distribution of peculiar velocities in all models,as expected from a
Gaussian initial density field. The cluster 3-D velocity distribution typically
peaks at $v \sim 600$ km s$^{-1}$, and extends to high cluster velocities of $v
\sim 2000$ km s$^{-1}$. The low-density CDM model exhibits somewhat lower
velocities: it peaks at $\sim 400$ km s$^{-1}$, and extends to $\sim 1200$ km
s$^{-1}$. Approximately 10\% ($\sim$1\% for low- density CDM) of all model rich
clusters move with high peculiar velocities of $v \ge 10^3$ km s$^{-1}$. The
highest velocity clusters frequently originate in dense superclusters. The
model velocity distributions of groups and clusters of galaxies are compared
with observations. The data exhibit a larger high-velocity tail, to $v_r \geq
2000$ km s$^{-1}$, than seen in the model simulations (except HDM). Due to the
large observational uncertainties, however, the data are consistent at a $\sim
1$ to $3\sigma$ level with the model predictions and with a Gaussian initial
density field. Accurate observations of cluster peculiar velocities, especially
at the high-velocity tail, should provide powerful constraints on the
cosmological models.
