With a chronological review of Hurwitz theory, we survey some known results on the enumeration of the equivalence classes of several types of branched coverings of a surface. In particular, relations with the enumeration of the equivalence classes of several types of graph coverings and enumerating the isomorphism classes of branched orientable surface coverings of a nonorientable surface will be mentioned. Also we discuss a similar problem for branched coverings having prescribed branched types.