The assumption of stationarity is appealing in classical time series analysis because of the ease of estimation and forecasting. However, stationarity is an idealization which, in practice, can at best hold as an approximation, but for many time series may be an unrealistic assumption. In this talk, we define a class of locally stationary processes which can lead to improved forecasts and more accurate uncertainty quantification. This class of processes assumes the model parameters to be time-varying and parameterizes them in terms of a transformation of basis functions that ensures that the processes are locally stationary. We investigate methods and theory for parameter estimation in this class of models, and propose tests of stationarity that have the potential to allow us to examine certain departures from stationarity in time series data. We assess our methods using simulation studies and apply these techniques to the analysis of climate time series.