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We study the H-regular surfaces, a class of intrinsic regular hypersurfaces in the setting of the Heisenberg group H^n = C^n x R = R^{2n+1} endowed with a left-invariant metric d equivalent to its Carnot-Carathéodory (CC) metric. Here hypersurface simply means topological codimension 1 surface and by the words "'intrinsic'" and "'regular" we mean, respectively, notions involving the group structure of H^n and its differential structure as CC manifold. In particular, we characterize these surfaces as intrinsic regular graphs inside H^n by studying the intrinsic regularity of the parameterizations and giving an area-type formula for their intrinsic surface measure

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In the framework of Carnot-Carathéodory spaces we study Minkowski content and perimeter, we prove some coarea formulas, and finally we prove some variational approximations of the perimeter.

We study the notion of intrinsic Lipschitz graphs within Heisenberg groups, focusing our attention on their Hausdorff dimension and on the almost everywhere existence of (geometrically defined) tangent subgroups. In particular, a Rademacher type theorem enables us to prove that previous definitions of rectifiable sets in Heisenberg groups are natural ones.