Spherical CR Geometry is the geometry carried by the 3-sphere when it is viewed as the ideal boundary of the complex hyperbolic plane, a 4-dimensional rank one symmetric space of negative curvature. A spherical CR structure on a 3-manifold is a geometric structure modeled on spherical CR geometry. For quite a long time it was not known if a 3-manifold could admit both a hyperbolic structure and a spherical CR structure. I will explain an analogue of Thurston's celebrated theorem in the context of spherical CR geometry. I will then how to use this result to construct an infinite list of pairwise incommensurable closed hyperbolic 3-manifolds which also admit spherical CR structures. The spherical CR surgery theorem also has applications to complex hyperbolic discrete groups: I will explain how to use it to solve the so-called (p,q,r)-Goldman-Parker conjecture for large parameters.