The large deviation approach to statistical mechanics explores the exponential decay of probabilities of large fluctuations in random systems. This theory is crucial in fields like statistics, finance, and engineering, as it provides insights into the behavior of systems around their most probable states. In equilibrium statistical mechanics, large deviation theory refines and generalizes Einstein's fluctuation theory. This review aims to show that the mathematical language of statistical mechanics is essentially that of large deviation theory. It covers the basics of large deviation theory, its applications to sums of random variables and Markov processes, and its connections to various physical systems, including equilibrium and nonequilibrium systems, multifractals, disordered systems, and chaotic systems. The review also discusses fundamental aspects of statistical mechanics, such as variational principles, the breaking of the Legendre transform for nonconcave entropies, and fluctuation relations. The text provides a detailed exploration of large deviation principles, rate functions, and their applications in statistical mechanics, emphasizing the mathematical foundations and their physical interpretations. It includes examples of large deviation results, such as the random bits and Gaussian sample mean, and discusses the Gärtner-Ellis Theorem and Cramér's Theorem. The review also addresses the properties of the scaled cumulant generating function and rate functions, highlighting their convexity and the Legendre transform relationship. The text concludes with an overview of the applications of large deviation theory in statistical mechanics, including equilibrium and nonequilibrium systems, and its broader implications in physics.The large deviation approach to statistical mechanics explores the exponential decay of probabilities of large fluctuations in random systems. This theory is crucial in fields like statistics, finance, and engineering, as it provides insights into the behavior of systems around their most probable states. In equilibrium statistical mechanics, large deviation theory refines and generalizes Einstein's fluctuation theory. This review aims to show that the mathematical language of statistical mechanics is essentially that of large deviation theory. It covers the basics of large deviation theory, its applications to sums of random variables and Markov processes, and its connections to various physical systems, including equilibrium and nonequilibrium systems, multifractals, disordered systems, and chaotic systems. The review also discusses fundamental aspects of statistical mechanics, such as variational principles, the breaking of the Legendre transform for nonconcave entropies, and fluctuation relations. The text provides a detailed exploration of large deviation principles, rate functions, and their applications in statistical mechanics, emphasizing the mathematical foundations and their physical interpretations. It includes examples of large deviation results, such as the random bits and Gaussian sample mean, and discusses the Gärtner-Ellis Theorem and Cramér's Theorem. The review also addresses the properties of the scaled cumulant generating function and rate functions, highlighting their convexity and the Legendre transform relationship. The text concludes with an overview of the applications of large deviation theory in statistical mechanics, including equilibrium and nonequilibrium systems, and its broader implications in physics.