Numerous results of many authors have established deep conections between the parabolic Allen-Cahn equation and motion by mean curvature. We will discuss the little that is known about similar relations between semilinear wave equations and a natural hyperbolic analogue of the mean curvature flow. In particular, we will review some older results and possibly relevant geometric measure theory constructions, and we will describe a new, constructive approach to these problems (joint work with M del Pino and M Musso).