  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 CDM, HDM ($\Omega=1$), and PBI. We find that rich clusters of
galaxies exhibit a Maxwellian distribution of peculiar velocities in all
models, as expected from a Gaussian initial density fluctuation field. The
cluster 3-D velocity distribution is generally similar in the models: it peaks
at $v \sim 500$ km s$^{-1}$, and extends to high cluster velocities of $v \sim
1500$ km s$^{-1}$. Approximately 10\% 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 group velocity distribution
is, in general, similar to the velocity distribution of the rich clusters. In
all but the low-density CDM model, the mass exhibits a longer tail of high
velocities than do the clusters. This high-velocity tail originates mostly from
the high velocities that exist within rich clusters. The model velocity
distributions of groups and clusters of galaxies are compared with
observations. The data are generally consistent with the models, but exhibit a
somewhat larger high-velocity tail, to $v_r \sim 3000$ km s$^{-1}$. While this
high-velocity tail is similar to the HDM model predictions, the data are
consistent with the other models studied, including the low-density CDM model,
which best fits most other large-scale structure observations. The observed
velocity
