Let F be a finite subgroup of the mapping class group of a closed surface S of genus greater than 2, and let N(F) be the normalizer of F in the mapping class group. The main result of this paper is to give a description of N(F) as an extension of F by an explicitly defined subgroup of finite index in the mapping class group of an associated quotient surface S/F , punctured at the branch points. The quotient group N(F)/F fits into an exact sequence NB -> N (F)/F -> NS , where NB is of finite index in a braid group and NS is of finite index in the mapping class group of S/F . A characterization of the image of N(F) in Aut(F) is given so that the centralizer, Z(F), is also determined. These results are then applied to the important case of elementary abelian p-subgroups of mapping class groups. Sample results include a classification of all elementary abelian subgroups with finite normalizers and a description of the normalizers of the rank 1 elementary abelian subgroups.