If a spacetime singularity is always covered within an event horizon of gravity, therefore hiding it from external observers in the Universe, we then have a firm basis to develop an entire black holes physics which can also be applied to other astrophysical scenarios. The assumption that no visible or naked singularity occurs during the collapse of a massive star was developed into a black hole theory in the 1970s. Such an assumption is called cosmic censorship conjecture, which is fundamental to black hole physics and its major astrophysical applications. The chapter discusses this conjecture here, as well as the various aspects of this hypothesis. It indicates the difficulties in proving the censorship, pointing out the need, therefore, to study gravitational collapse models in general relativity. The chapter specifies what a naked singularity is, and describe its main characteristics in gravitation theory. The consideration of inhomogeneous collapse provides much insight into the black hole and singularity formation and this work of recent years is reviewed and discussed here.
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