This paper is based on the Wald Lectures diven at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations. The author has considered following sections of the theory: 1. Large deviations for sums; 2. Rate functions, duality and generating functions; 3. Markov processes; 4. Small random perturbations. The exit problem; 5. Gibbs measures and statistical mechanics; 6. Interacting particle systems; 7. Superexponential estimates; 8. Hydrodynamical limits; 9. Large deviations in hydrodynamical limits; 10. Large deviations for random walks in random environments; 11. Homogenization of Hamilton-Jacobi-Bellman equations; 12. History and references. Remark: From the revierwer's point of view, a survey on the modern theory of large deviations should also include 1) the classical paper by \textit{A. A. Borovkov} [Th. Probab. Appl., 12, 575--595 (1967; Zbl 0178.20004)], which together with the papers by \textit{I. N. Sanov} [Ref. 18, Mat. Sb., N. Ser. 42(84), 11--44 (1957; Zbl 0078.31202)] and by \textit{S. R. S. Varadhan} [Ref. 20, Commun. Pure Appl. Math. 20, 659--685 (1967; Zbl 0278.60051)] lays in the basis of modern theory of large deviations; 2) the paper by \textit{A. A. Puhalskii} [Theory Probab. Appl. 38, 490--497 (1993; Zbl 0807.60036)], in which necessary and sufficient conditions for large deviation principle are suggested.