Universal (Co) Homology Theories

It is shown that the homology and cohomology theories on separable C^*-algebras given by nonstable E-theory are the universal such theories. By specializing to Abelian C^*-algebras, we obtain a family of extraordinary Steenrod homology and cohomology theories on pointed compact metric spaces which are the universal such theories in the same way. For each of the extraordinary Steenrod (co)homology theories considered, we describe an Ω-spectrum which represents the theory.