A previously derived class of solutions representing gravitationally collapsing dust clouds is investigated in detail. The class includes all spherically symmetric collapses but is more general and is termed quasispherical. Most features of spherical collapse appear to go over in their essentials to the more general class. Under reasonable initial conditions the singularity is hidden behind an event horizon, but with slightly irregular initial density distributions portions of the singularity become naked. The latter feature is not restricted to the nonspherical case and will in general appear in situations such as collapsing spherical shells of matter. (AIP)