The $n$-dimensional matrix representations of a group or an algebra $A$ form a space $\mathrm{Rep}(A,n)$ called a representation variety of $A$. This is a classical geometric invariant that plays a role in many areas of mathematics. The construction $\mathrm{Rep}(A,n)$ is natural in $A$, but not 'exact' in the sense of homological algebra. We'll explain how to refine $\mathrm{Rep}(A,n)$ by constructing a derived representation variety $\mathrm{DRep}(A,n)$, an example of a derived moduli space in algebraic geometry. For an application, we'll look at the varieties of commuting matrices and present some combinatorial conjectures extending the famous Macdonald conjectures in representation theory.