The singularity theorems of Hawking and Penrose tell us that singularities are common place in general relativity. Singularities not only occur at the beginning of the Universe at the Big Bang, but also in complete gravitational collapses that result in the formation of black holes. If singularities - except the one at the Big Bang - ever become "naked," i.e. not shrouded by black hole horizons, then it is expected that problems would arise and render general relativity indeterministic. For this reason, Penrose proposed the cosmic censorship conjecture, which states that singularities should never be naked. Various counterexamples to the conjecture have since been discovered, but it is still not clear under which kind of physical processes one can expect violation of the conjecture. In this short review, I briefly examine some progresses in space-time singularities and cosmic censorship conjecture. In particular, I shall discuss why we should still care about the conjecture, and whether we should be worried about some of the counterexamples. This is not meant to be a comprehensive review, but rather to give an introduction to the subject, which has recently seen an increase of interest.
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