A source of many interesting and deep complications in C*-algebras is the fact that taking tensor products does not always preserve exact sequences or inclusions. Similar problems arise when taking crossed products by group actions. There are several universal constructions of tensor-product functors crossed-product functors with prescribed exactness, injectivity or Morita-invariance properties. After motivating and introducing these functors, I will report on some recent progress on the interrelations between them. This is based on joint work with Timo Siebenand.