We introduce and study the evolution by horizontal mean curvature flow in Carnot-Carathe'odory spaces. The main difficulty occurs because of the existence of characteristic points, which are points where the horizontal normal is not defined even if the Euclidean normal exists. The existence of these points make the evolution by horizontal mean curvature flow very different from the corresponding Euclidean or Riemannian evolution. We introduce a definition corresponding to the Euclidean generalized evolution by mean curvature flow for the level sets. We will present all the results known so far for this geometric evolution and in particular a stochastic approach which was first introduced by Buckdahn-Cardaliaguet-Quincampoix (2001) and Soner-Touzi, (2002) in the analogous Euclidean case. We will use this approach in order to get existence for the generalized evolution by horizontal mean curvature flow in any sub-Riemannian geometries by the definition of suitable intrinsic controlled Brownian motions. Joint work with N. Dirr, University of Cardiff and M. von Renesse, University of Leipzig.