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 ($��=1$) CDM, low-density CDM, HDM ($��=1$), 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

25p plaintex submitted to The Astrophysical Journal