Peter McCullagh has made wide-ranging contributions to statistical theory and methods. His book, Tensor Methods in Statistics (1987), is a pioneering exploitation of tensor methods applied to distributional problems arising in statistics. It contains substantial new results, in particular on the distribution of likelihood-based statistics widely used in applications. His book with John A. Nelder, Generalized Linear Models, (first edition 1983; second edition 1989) has had a major direct impact on statistical methods used in many fields of application; the second edition contains much new material developed since the first edition. Other work includes the elucidation of quasi-likelihood, a device for relaxing assumptions about errors made in many commonly used methods of statistical analysis, and the development of powerful methods for the analysis of ordinal data.