This chapter narrates the history of the furious debate that took place over the existence of spacetime singularities. After describing what a spacetime singularity is and pointing out its existence, we consider various options as to whether we can avoid such singularities. It is concluded that under a wide set of physically reasonable conditions, general relativity necessarily implies the occurrence of spacetime singularities when massive stars collapse in the Universe, and in cosmology at the origin of the Universe, otherwise known as the Big Bang. The chapter considers the issues of causality violations, energy conditions, and also the formation of trapped surfaces, which play key role in the occurrence of singularities. The basic physical challenges posed by spacetime singularities are narrated.
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