  In the present paper we consider generic Sub-Riemannian structures on the
co-rank 1 non-holonomic vector distributions and introduce the associated
canonical volume and ''horizontal'' area forms. As in the classical case, the
Sub-Riemannian minimal surfaces can be defined as the critical points of the
'`horizontal'' area functional. We derive an intrinsic equation for minimal
surfaces associated to a generic Sub-Riemannian structure of co-rank 1 in terms
of the canonical volume form and the ``horizontal'' normal. The presented
construction permits to describe the Sub-Riemannian minimal surfaces in a
generic Sub-Riemannian manifold and can be easily generalized to the case of
non-holonomic vector distributions of greater co-rank.
  The case of contact vector distributions, in particular the $(2,3)$-case, is
studied more in detail. In the latter case the geometry of the Sub-Riemannian
minimal surfaces is determined by the structure of their characteristic points
(i.e., the points where the hyper-surface touches the horizontal distribution)
and characteristic curves. It turns out that the known classification of the
characteristic points of the Sub-Riemannian minimal surfaces in the Heisenberg
group $H^1$ holds true for the minimal surfaces associated to a generic contact
$(2,3)$ distribution. Moreover, we show that in the $(2,3)$ case the
Sub-Riemannian minimal surfaces are the integral surfaces of a certain system
of ODE in the extended state space. In some particular cases the Cauchy problem
for this system can be solved explicitly. We illustrate our results considering
Sub-Riemannian minimal surfaces in the Heisenberg group and the group of
roto-translations.
