The success of gauge theory descriptions of Nature follows simply, in hindsight, from Lorentz symmetry, quantum mechanics, and the existence of interacting massless particles with spin. Yet, remarkably, the most generic type of massless particle spin has never been seriously examined: Wigner's so-called "continuous spin" particles (CSPs), which have a tower of polarization states, carrying all integer or half-integer helicities, that mix under boosts. I will explain recent progress in understanding these particles on two fronts: simple scattering amplitudes and a free quantum field theory. The scattering amplitudes give two remarkable insights into CSP physics. First, Lorentz symmetry protects CSP interactions from the dysfunction one might expect in a theory with infinitely many polarization states: divergent cross-sections and problematic thermodynamics. Second, and most intriguingly, CSP interactions approach those of ordinary scalars or helicity-1 or 2 gauge bosons in a high-energy "correspondence" regime. The free field theory has a simple geometric interpretation and recovers a sum of integer-helicity actions in a related correspondence limit. While a full interacting theory of CSPs remains elusive, these results suggest that any such theory would extend Maxwell electrodynamics and/or general relativity in a viable and testable way.