In evolution equations for interfaces, topological changes and geometric singularities often occur naturally, one basic example being the pinchoff of liquid droplets. As a consequence, classical solution concepts for such PDEs are naturally limited to short-time existence results or particular initial configurations like perturbations of a steady state. At the same time, the transition from strong to weak solution concepts for PDEs is prone to incurring unphysical non-uniqueness of solutions. We will first review some classical results and conjectures concerning the uniqueness properties for two particular interface evolution problems, the motion of an interface by its mean curvature and the motion of the interface between two immiscible fluids. We will then present weak-strong uniqueness principles for multiphase mean curvature flow as well as for the evolution of fluid-fluid interfaces: As long as a classical solution to these evolution problems exists, it is also the unique weak solution.