We have used the Hubble Space Telescope (HST) WFPC2 camera to survey 132 BL Lac objects comprising seven complete radio-, X-ray-, and optically selected samples. We obtained useful images for 110 targets spanning the redshift range 0<~z<~1.3. These represent an unbiased subsample of the original 132 since they were snapshots selected to fill random holes in the HST schedule. The exposure times ranged from a few hundred to ~1000 s, increasing with redshift. Most images were taken in the F702W filter; those already observed in F814W during Cycle 5 were reobserved in F606W to give broader wavelength coverage. The data were analyzed uniformly, and both statistical and systematic errors were estimated (the latter dominate). In 2/3 of the BL Lac images, host galaxies are detected, including nearly all for z<0.5 (58 of 63). In contrast, only one-quarter of the BL Lac objects with z>0.5 (six of 22) were resolved because of the relatively short exposure times, and these tend to be very luminous host galaxies. The highest redshift host galaxy detected is in a BL Lac object at z=0.664. HST data add critical morphological information in the range a few tenths to a few arcseconds. In 58 of the 72 resolved host galaxies, a de Vaucouleurs profile is significantly preferred, at >~99% confidence, over a pure exponential disk; the two fits are comparable in the remaining 14 cases because of their generally lower signal-to-noise ratios. These results limit the number of disk systems to at most 8% of BL Lac objects (at 99% confidence) and are consistent with all BL Lac host galaxies being ellipticals. The detected host galaxies are luminous ellipticals with a median absolute K-corrected magnitude of MR~-23.7+/-0.6 mag (rms dispersion), at least 1 mag brighter than M* and comparable to brightest cluster galaxies. The galaxy morphologies are generally smooth and undisturbed, with small or negligible ellipticities (ɛ<~0.2). The half-light surface brightness is anticorrelated with half-light radius in quantitatively the same way as other elliptical galaxies, indicating that apart from their highly active nuclei, BL Lac objects appear to be absolutely normal ellipticals. There is no correlation between host galaxy and observed nuclear magnitude or estimated jet power corrected for beaming. If black hole mass is correlated linearly with bulge mass in general, this implies a large range in Eddington ratio. The host galaxies of the radio-selected and X-ray-selected BL Lac objects are comparable in both morphology and luminosity, strongly suggesting that nuclear properties do not have a dramatic effect on large-scale host galaxy properties, or vice versa. BL Lac objects have extended radio powers and host galaxy magnitudes very much like those of FR I galaxies, and quite distinct from FR II's, which instead are more similar to quasars. Thus the present data strongly support the unification picture with FR I galaxies constituting the bulk of the parent population of BL Lac objects.