Equivariant homologies for operator algebras

This is a survey of a variety of equivariant (co)homology theories for operator algebras. We briefly discuss a background on equivariant theories, such as equivariant $K$-theory and equivariant cyclic homology. As the main focus, we discuss a notion of equivariant $L^2$-cohomology and equivariant $L^2$-Betti numbers for subalgebras of a von Neumann algebra. For graded $C^*$-algebras (with grading over a group) we elaborate on a notion of graded $L^2$-cohomology and its relation to equivariant $L^2$-cohomology.