After a brief introduction to Einstein’s theory of general relativity and its most profound prediction of black holes, I will focus on spacetime singularities, i.e., regions where general relativity breaks down and must be replaced by a quantum theory of gravity.  I first discuss singularities inside black holes. This is the usual case and is an old story, but there have been some recent developments. I will next describe some new results which show that some black holes have singularities on their surface. Finally, I will discuss the possibility of singularities outside black holes.