We present Hubble Space Telescope (HST) WF/PC and WFPC2 observations of four rich clusters of galaxies at redshifts z ~ 0.4: CL 1447+2619, CL 0939+4713, CL 0024+1654, and Abell 370. Extensive ground-based observations have shown that all have substantial populations of blue, star-forming, and (in some cases) starbursting galaxies. We find the galaxy populations in these clusters to be approximately normal, in that most galaxies can be accommodated in the z = 0 Hubble sequence and have the luminosities, colors, and distributions within the clusters that are expected for their Hubble types. However, in every cluster environment the galaxy populations contain a uniformly higher proportion of late-type galaxies than is found in similar environments today.Confirming previous ground-based and HST observations, we find a remarkably large fraction of disturbed systems. The appearance of many of these is suggestive of merging and tidally interacting objects, but several difficulties with this suggest that the newly described process of ``galaxy harassment'' may be a more likely explanation. These objects are clearly responsible for the majority of the starburst spectra detected in ground-based observations of the clusters. They cannot, however, account for most of the large population of blue galaxies. The fraction of such disturbed galaxies varies from cluster to cluster in concert with the number of blue and of spiral galaxies. All these variations are well correlated with the structure and density of the clusters, but not with cluster richness. Close examination of the population of ``normal'' spirals shows that a significant fraction have very abnormal patterns of star formation: Rings of star formation are much more common than two-armed spirals. Galaxies of this type are very rare in nearby populations but also have been found in HST observations of high-redshift field populations. We speculate that these forms might be products of the cluster environment or may simply represent the norm for the earlier stage of disk evolution observed at this redshift. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.