We present in this paper a novel approach to the long-standing problem of motion planning for non-holonomic systems. Our method is built upon a parabolic partial differential equation that arises in the study of Riemannian manifold. We show how it can be brought to bear to provide a solution to a non-holonomic motion planning problem. We illustrate the method on canonical examples, namely the unicycle, the non-holonomic integrator, and the parallel parking task for a non-holonomic car model. We also brie#y address computational issues pertinent to solving this particular partial differential equation, and point out the existence of fast algorithms and the fact that the problem is easily parallelizable.