We consider group actions a:G->A of topological groups on C*-algebras A of the type which occur in many quantum physics models. These are singular actions in the sense that they need not be strongly continuous, or the group need not be locally compact. We develop a "crossed product host" in analogy to the usual crossed product for strongly continuous actions of locally compact groups, in the sense that its representation theory is in a natural bijection with the covariant representation theory of the action a:G->A. We have a uniqueness theorem for crossed product hosts, useful existence conditions, and a number of examples where a crossed product host exists, but the usual crossed product does not. In the case that the class of permissible covariant representations is restricted by a positive spectral condition, there is additional theory available for construction of crossed product hosts, and if time permits, we will discuss some results in this area.