When does a Boltzmannian equilibrium exist?

We present a definition of equilibrium for Boltzmannian statistical mechanics based on the long-run fraction of time a system spends in a state. We then formulate and prove an existence theorem which provides general criteria for the existence of an equilibrium state. We illustrate how the theorem works with toy example. After a look at the ergodic programme, we discuss equilibria in a number of different gas systems: the ideal gas, the dilute gas, the Kac gas, the stadium gas, the mushroom gas and the multi-mushroom gas.