Consider a relative two-complex (L,K) where K is an aspherical CW complex (of arbitrary dimension). The pair (L,K) is said to be “aspherical" if the second relative homotopy group of the pair (L,K) is trivial, which is equivalent to saying that the inclusion of K in L induces a monomorphism of fundamental groups and that the complex L is aspherical in the traditional sense. In this case one can deduce a lot of group theoretic information about the fundamental group of L in terms of that of K. In this talk, I will consider the alternatives to asphericity, examining in detail the case where the quotient complex L/K is "dunce cap" modeled on the presentation (x:xxx^-1). Asphericity was classified in this setting by M. Edjvet in 1993. I will survey the non-aspherical cases and conclude that except for a small handful of unresolved cases, if (L,K) is not aspherical, then the fundamental group of L either contains a "forbidden" finite subgroup or else splits as an amalgamated free product over a virtual three-manifold group. Some interesting and well-known finite groups and three-manifolds occur in this survey.