The past decade has seen a number of advances in our understanding of how phases of matter can be distinguished by phenomena other than spontaneous symmetry breaking, notably through global topological properties, such as emergent quasiparticles with fractional statistics known as anyons, or symmetry-protected gapless boundary states. A particularly rich set of possibilities comes from combining these two, and studying the different possible ways that unbroken symmetries can act on systems of anyons. I will explore several ways that symmetry can act in anyon systems, and show how studying these deepens our understanding of a number of other phenomena, from symmetry-protected surface states of 3 dimensional systems to the phase structure of 2D anyon systems.