Equivariant homologies for operator algebras; a survey

‎This is a survey of a variety of equivariant (co)homology theories for operator algebras‎. ‎We briefly discuss a background on equivariant Hochschild cohomology‎. ‎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.