Suppose we observe the sum of two structured signals, and we must identify the components in the mixture. This setup includes separating two signals that are sparse in different bases and the problem of separating a sparse matrix from a low-rank matrix. This talk describes a convex optimization framework for solving these demixing problems and others. We present a randomized signal model that encapsulates the notion of ``incoherence'' between two structures. For this model, spherical integral geometry provides exact formulas that describe when the optimization problem will succeed (or fail) to demix the component signals with high probability. We argue that our ability to separate two structured signals depends only on the total ``complexity'' of the two structures.