  A hollow vortex is a region of constant pressure bounded by a vortex sheet
and suspended inside a perfect fluid; it can therefore be interpreted as a
spinning bubble of air in water. This paper gives a general method for
desingularizing non-degenerate steady point vortex configurations into
collections of steady hollow vortices. Our machinery simultaneously treats the
translating, rotating, and stationary regimes. Through global bifurcation
theory, we further obtain maximal curves of solutions that continue until the
onset of a singularity. As specific examples, we give the first existence
theory for co-rotating hollow vortex pairs and stationary hollow vortex
tripoles, as well as a new construction of Pocklington's classical
co-translating hollow vortex pairs. All of these families extend into the
non-perturbative regime, and we obtain a rather complete characterization of
the limiting behavior along the global bifurcation curve.
