The skill of a chess player is measured by his or her Elo rating. We use a hierarchical Bayesian model to investigate the trajectory of a chess player's Elo rating across his or her career. Based on exploratory data analysis, we use a location-scale t-distribution to model the change in a player's rating from one year to the next. The parameters of the t-distribution are functions of the player's current age and rating. The model is autoregressive: the changes in a player's rating over the previous several years influence the location of the distribution for the next year. The model parameters are estimated via Markov chain Monte Carlo methods. We assess the performance of the model by calculating coverage percentages of interval forecasts for a large dataset of players compiled by the chess organization FIDE. We also use the model to give probabilistic forecasts for the rating trajectories of several top chess players and prodigies, and to investigate the age at which players typically achieve their peak ratings.