Almost every physical theory involves differential equations, and conversely, the mathematical theory of differential equations gets much of its intuition from physics. The physical observation that nature determines the future if we just supply the initial conditions has its counterpart in the theory of differential equations: existence and uniqueness theorems for solutions. This intuition is so familiar that it may come as a surprise (it did to me) to learn that solutions may fail to be unique. I will describe how this can happen, and two physical situations where it actually does happen, in an appropriate singular limit. These two situations, Laplacian growth and turbulence, are outstanding problems which are still, in many ways, mysterious. One cannot help wondering if the failure of uniqueness is perhaps trying to tell us something. But what? The question is largely unexplored.