A spherical CR structure on a 3-manifold is a system of coordinate charts into the 3-sphere, such that the overlap functions are restrictions of complex projective automorphisms. I will discuss an analogue of Thurston's celebrated hyperbolic Dehn surgery theorem in the context of spherical CR geometry. I'll explain how the spherical CR surgery result constructs many examples of spherical CR structures on closed manifolds - including closed hyperbolic manifolds. I'll also explain, to some extent, how the CR surgery theorem sheds light on the structure of representations of triangle groups into PU(2,1), the group of complex projective automorphisms of the 3-sphere.