It has been proposed, as a measure of the complexity of a projective variety, to study its associated category of arithmetically Cohen-Macaulay (aCM) sheaves. When there exist families of arbitrarily large dimension of indecomposable aCM sheaves, we say that the variety is of wild representation type. Most of the research in this area has been done under the assumption that the underlying variety is also aCM. Indeed, jointly with D. Faenzi, we proved that any reduced aCM variety is of wild representation type, except for a few cases which are completely classified. In this talk I am going to talk about these topics (which involve the theory of Ulrich bundles) as well as some recent results on the wildness of a broad class of non-aCM