We report first results from a series of N-body/gasdynamical simulations designed to study the origin of galaxy morphologies in a cold dark matter-dominated universe. The simulations include star formation and feedback and have numerical resolution sufficiently high to allow for a direct investigation of the morphology of simulated galaxies. We find, in agreement with previous theoretical work, that the presence of the main morphological components of galaxies—disks, spheroids, bars—is regulated by the mode of gas accretion and intimately linked to discrete accretion events. In the case we present, disks arise from the smooth deposition of cooled gas at the center of dark halos, spheroids result from the stirring of preexisting disks during mergers, and bars are triggered by tides generated by satellites. This demonstrates that morphology is a transient phenomenon within the lifetime of a galaxy and that the Hubble sequence reflects the varied accretion histories of galaxies in hierarchical formation scenarios. In particular, we demonstrate directly that disk/bulge systems can be built and rebuilt by the smooth accretion of gas onto the remnant of a major merger and that the present-day remnants of late dissipative mergers between disks are spheroidal stellar systems with structure resembling that of field ellipticals. The perplexing variety of galaxy morphologies is thus highly suggestive of—and may actually even demand—a universe where structures have evolved hierarchically.