A general theme in geometry is the search for connections between the topo- logical properties of a space and the geometrical properties of finer struc- tures on that space. Thurston’s hyperbolic Dehn surgery theorem (see [T0]) is a great theorem along these lines: All but finitely many Dehn fillings performed on a cusp of a hyperbolic 3-manifold result in new hyperbolic 3-manifolds. See Section 1.1. The purpose of this monograph is to prove an analogue of Thurston’s result in the setting of spherical CR geometry and then to derive some consequences from it. We call our result the Horotube Surgery Theorem, or HST for short. See Theorem 1.2.