This book considers periodic time series models for seasonal data, characterized by parameters that differ across the seasons, and focuses on their usefulness for out-of-sample forecasting. Providing an up-to-date survey of the recent developments in periodic time series, the book presents a large number of empirical results. The first part of the book deals with model selection, diagnostic checking, and forecasting of univariate periodic autoregressive models. It discusses tests for periodic integration, and provides an extensive discussion of the role of deterministic regressors in testing for periodic integration and in forecasting. The second part discusses multivariate periodic autoregressive models. It provides an overview of periodic cointegration models, including single-equation type tests and a full-system approach based on generalized method of moments. All methods are illustrated with extensive examples, and the book will be invaluable to advanced graduate students and researchers in econometrics and to practitioners looking for an understanding of how to approach seasonal data.