This book provides one of the first accounts of statistical analysis and inference based on approximate models Developed by the author, this approach consistently treats models as approximations to data, not to some underlying truth. The author develops a concept of approximation for probability models with applications to discrete data, location scale, ANOVA, nonparametric regression, image analysis, densities, time series, and model choice. He also offers a critique of statistics that covers likelihood, Bayesian statistics, sufficient statistics, efficiency, asymptotics, and model choice.