A regular epimorphism in an arbitrary category is a morphism which is a coequalizer of some pair of parallel morphisms. They are usually seen as “good” epimorphisms but their concrete description is often a problem. This paper provides a description of regular epimorphisms in the category of topological spaces and in some other similar categories, in terms of convergence of ultrafiters. This result is a special instance of a more general one on regular epimorphisms of \((T; \,V)\)-algebras where \(V\) is a symmetric monoidal closed complete lattice and \(T\) is a \(V\)-admissible monad on \({\mathcal S}et\).