Abstract: The Balmer spectrum of a tensor triangulated category is a topological tool analogous to the usual spectrum of a commutative ring. It provides a universal theory of support, giving a categorical framework to (among others) the support varieties that have been used to great effect in modular representation theory. In this talk I will give a brief overview of the support theory for Hochschild cohomology and present the basic definitions and properties of support varieties in relative Hochschild cohomology. This will include an unpretentious introduction to the Balmer spectrum and its appearances in representation theory, as well as the basic notions of relative homological algebra. The talk will conclude with reasons to be hopeful for applications to Lie superalgebras.