ABSTRACT. We consider the sub-Riemannian metric g h on S 3 provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot-Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus.We study area-stationary surfaces with or without a volume constraint in (S 3 , g h ). By following the ideas and techniques in [RR2] we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volumepreserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot-Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C 2 compact, connected, embedded surfaces in (S 3 , g h ) with empty singular set and constant mean curvature H such that H/ √ 1 + H 2 is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (S 3 , g h ).