  The essential singularity in Einstein's gravity can be avoidable if the
preconditions of Penrose's theorem can be bypassed, i.e., if the strong energy
condition is broken in the vicinity of a black hole center. The singularity
mentioned here includes two aspects: (i) the divergence of curvature
invariants, and (ii) the incompleteness of geodesics. Both aspects are now
taken into account in order to determine whether a black hole contains
essential singularities. In this sense, black holes without essential
singularities are dubbed regular (non-singular) black holes. The regular black
holes have some intriguing phenomena that are different from those of singular
black holes, and such phenomena have inspired numerous studies. In this review,
we summarize the current topics that are associated with regular black holes.
