Time series models are often constructed by combining nonstationary effects such as trends with stochastic processes that are known (or believed) to be stationary. However, there are numerous time series models for which the stationarity of the underlying process is conjectured but not yet proven. We give an approachable introduction to the use of drift criteria (also known as Lyapunov function techniques) for establishing strict stationarity and ergodicity of such models. These conditions immediately imply consistent estimation of the mean and lagged covariances, and more generally the expectation of any integrable function. We demonstrate by proving stationarity and ergodicity for several novel and useful examples, including Poisson log-link Generalized Autoregressive Moving Average models.