Maximum principles make it possible to bound the values of a certain function by its values on a small subset of its domain. In particular, they occur in potential theory, where one seeks to bound the potential of a measure by its values on the support of the measure. Applications of maximum principles include the characterization of Carleson measures for certain spaces of analytic functions. I will talk about a new approach to maximum principles in potential theory, connecting them to the linear algebraic notion of copositivity. This is joint work in progress with Nikolaos Chalmoukis.