Recent measurements on 2d materials tuning between fractional quantum anomalous Hall phases and a plethora of correlated electronic states call for a detailed understanding of the dynamics of anyons. In this talk, I will develop a general theory of the statistical mechanics of anyon gases, valid in regimes where the anyons are sufficiently dilute. With a minimal set of universal braiding and fusion data, along with information about the hierarchy of anyon gaps, I will discuss how it is possible to construct a distribution function for any dilute anyon gas, as well as derive thermodynamic observables. Our results are built on a new kind of anyon exclusion principle, which arises as a constraint on fusion outcomes of physical states, complementing earlier proposals of fractional exclusion statistics. Our approach unifies and streamlines a range of results for itinerant anyon models, from solvable lattice Hamiltonians to large-N field theories.