The chapter introduces the theory of large deviations, which focuses on the exponential decay of probabilities for large fluctuations in random systems. This theory is crucial in various fields, including statistics, finance, and engineering, as it provides valuable insights into the behavior of systems around their most probable states or trajectories. In the context of equilibrium statistical mechanics, large deviation theory offers refined and generalized estimates of probabilities, extending Einstein's theory of fluctuations. The review explores the connections between large deviation theory and statistical mechanics, demonstrating that the mathematical language of statistical mechanics is essentially the language of large deviation theory. The first part of the review covers the basics of large deviation theory, including classical applications related to sums of random variables and Markov processes. The second part delves into various problems and results in statistical mechanics, showing how they can be formulated and derived within the framework of large deviation theory. These problems and results cover a wide range of physical systems, such as equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, multifractals, disordered systems, and chaotic systems. Key topics include the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations. The review also discusses fundamental aspects of statistical mechanics, such as the relationship between entropy and free energy functions, and the Legendre transform connecting them. The chapter emphasizes the importance of large deviation theory in understanding the behavior of complex systems and highlights its applications in both equilibrium and nonequilibrium statistical mechanics. It concludes by arguing that large deviation theory and statistical mechanics share a deep connection, with the mathematics of statistical mechanics being essentially the theory of large deviations.The chapter introduces the theory of large deviations, which focuses on the exponential decay of probabilities for large fluctuations in random systems. This theory is crucial in various fields, including statistics, finance, and engineering, as it provides valuable insights into the behavior of systems around their most probable states or trajectories. In the context of equilibrium statistical mechanics, large deviation theory offers refined and generalized estimates of probabilities, extending Einstein's theory of fluctuations. The review explores the connections between large deviation theory and statistical mechanics, demonstrating that the mathematical language of statistical mechanics is essentially the language of large deviation theory. The first part of the review covers the basics of large deviation theory, including classical applications related to sums of random variables and Markov processes. The second part delves into various problems and results in statistical mechanics, showing how they can be formulated and derived within the framework of large deviation theory. These problems and results cover a wide range of physical systems, such as equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, multifractals, disordered systems, and chaotic systems. Key topics include the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations. The review also discusses fundamental aspects of statistical mechanics, such as the relationship between entropy and free energy functions, and the Legendre transform connecting them. The chapter emphasizes the importance of large deviation theory in understanding the behavior of complex systems and highlights its applications in both equilibrium and nonequilibrium statistical mechanics. It concludes by arguing that large deviation theory and statistical mechanics share a deep connection, with the mathematics of statistical mechanics being essentially the theory of large deviations.