  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.
