Hill's spherical vortex is a classical explicit solution to the three dimensional Euler equations. It is an axisymmetric and swirl-free traveling wave solution with compactly supported yet discontinuous vorticity. In this talk we will explore the space of nearby Euler steady states and, in particular, prove that Hill's solution is the limit of traveling vortex rings that are smooth to infinite order. Along the way, we shall unveil certain pathological aspects of this space of steady solutions. This is forthcoming joint work with Razvan Radu.