A gravitational collapse inevitably results in a spacetime singularity, once the collapse has gone beyond a certain point, according to general relativity theory. The phrase {\it cosmic censorship} refers to two closely related conjectures about the nature of these spacetime singularities, due to Roger Penrose. The {\it weak} cosmic censorship conjecture states in essence says that gravitational collapse from regular initial conditions never creates a spacetime singularity visible to distant observers. The idea here is that any singularity that forms must be hidden within a black hole. The {\it strong} cosmic censorship conjecture holds that any such singularity is never visible to any observer at all, even someone close to it. A {\it naked} spacetime singularity would therefore be one that contradicts these conjectures. This article summarize some recent progress on these questions, and also on some related issues in general relativity theory. If naked singularities could be produced in our universe, astrophysical consequences might be intriguing. This possibility seems remote, however, and furthermore, no general theory of naked singularities exists at present, and no reliable ``fingerprints" of a naked singularity are known. This contrasts greatly with the situation on black holes, where the general theory has been soundly based since the early 1970's, and where possible observational signs are reasonably well understood. There is one seemingly naked singularity obvious to everyone, the initial singularity of the Big Bang. This singularity is not really a counterexample to the conjectures, because it did not form from regular initial conditions --- and could not have done so according to the singularity theorems of general relativity theory. This report also discuss the sense in which the Big Bang singularity can be regarded as hidden, not within a black hole, but behind a barrier of infinite cosmological redshift.
A new result is given on gravitational collapse: A theorem is announced about the existence of an apparent horizon in general relativity, which applies equally well to vacuum configurations and matter configurations. The theorem uses the reciprocal of the surface-to-volume ratio of a region on a space slice to measure the radius of the region, and uses the minimum value $K_{\rm min}$ of certain components of the extrinsic curvature to measure the strengh of the gravitational field in the region. The theorem proves that, if the product of the radius times $K_{\rm min}$ is larger than unity, then an apparent horizon must form, signalling the formation of a black hole.
Another new result is given: A number of people have shown that unitarity suffers a breakdown in quantum field theory, when one tries to turn on interactions in a causality-violating background (in a ``time machine"). I approach this problem by constructing a different free quantum field theory on a causality-violating background, by giving up the assumption that pure states persist. On the contrary, I allow for a model ``time machine" to evolve pure states into mixed states in a unique and well defined manner. This formulation of field quantum field theory still has a good classical limit when one restricts to coherent states, which do remain pure and coherent, and do not evolve into mixed states. I briefly summarize a way to preserve conservation of probability of interacting quantum field theories in this formulation.