In Lorentz-invariant theories, the polarization states of massless particles can in principle mix covariantly under boosts, historically referred to as "continuous spin". Exactly boost-invariant helicity -- which we are used to assuming -- is a special case, and corresponds to a vanishing spin Casimir, rho=0. But the value of rho need not vanish, and when non-zero, the physics differs from familiar expectations in an interesting and smooth manner. In this talk, I will introduce the core elements of a gauge theory formalism describing massless particles with nonzero rho, as well as basic aspects of their interactions with matter. As an illustration of the physics in a simple yet relevant example, I'll describe classical and quantum electromagnetism with nonzero rho, elementary aspects of gravity with nonzero rho, and highlight a number of specific ρ-dependent predictions to simple observables. I will also discuss ideas for how this physics might relate to dark matter, cosmology, and other aspects of the Standard Model.