  This paper considers a classical question of approximation of Brownian motion
by a random walk in the setting of a sub-Riemannian manifold $M$. To construct
such a random walk we first address several issues related to the degeneracy of
such a manifold. In particular, we define a family of sub-Laplacian operators
naturally connected to the geometry of the underlining manifold. In the case
when $M$ is a Riemannian (non-degenerate) manifold, we recover the
Laplace-Beltrami operator. We then construct the corresponding random walk, and
under standard assumptions on the sub-Laplacian and $M$ we show that this
random walk weakly converges to a process, horizontal Brownian motion, whose
infinitesimal generator is the sub-Laplacian. An example of the Heisenberg
group equipped with a standard sub-Riemannian metric is considered in detail,
in which case the sub-Laplacian we introduced is shown to be the sum of squares
(H\"{o}rmander's) operator.
