The mapping class group of a surface has an incredibly rich normal subgroup structure. For this reason, a traditional classification theorem for normal subgroups of mapping class groups in the form of a complete list of isomorphism types is almost certainly out of reach. In this talk, we will discuss recent work with Margalit giving an easy-to-check criterion for a normal subgroup to have the extended mapping class group as its automorphism group. This result applies to a wide class of normal subgroups and enables us to use the automorphism group as an invariant of normal subgroups, suggesting a new framework for the classification of normal subgroups of mapping class groups.