Given a finitely generated module \(M\) over a noetherian local ring \(R\), one may assign to it a conical affine variety, called the cohomological support variety of \(M\) over \(R\). This theory was first developed by Luchezar Avramov for local complete intersection rings in 1989, and by the work of many has recently been extended to encompass all commutative noetherian local rings. Geometric properties of this variety encode important homological information about \(M\) as well as \(R\). In this talk I will discuss what cohomological support varieties are, why they are useful, and some recent work on how they behave when restricting along a local homomorphism.