A sub-Riemannian manifold $M$ is a connected smooth manifold such that the only smooth curves in $M$ which are admissible are those whose tangent vectors at any point are restricted to a particular subset of all possible tangent vectors. Such spaces have several applications in physics and engineering, as well as in the study of hypo-elliptic operators. In this talk, we will construct a family of geometrically natural sub-elliptic Laplacian operators and discuss the trouble with defining one which is canonical. We will also construct a random walk on $M$ which converges weakly to a process whose infinitesimal generator is one of our sub-elliptic Laplacian operators. This is joint work with Masha Gordina.