The theory of large deviations is concerned with the exponential decay of probabilities of large
fluctuations in random systems. These probabilities are important in many fields of study, including
statistics, finance, and engineering, as they yield valuable information about the large fluctuations of
a random system around its most probable state or trajectory. In the context of equilibrium statistical
mechanics, the theory of large deviations provides exponential-order estimates of probabilities that
refine and generalize Einstein’s theory of fluctuations. This review explores this and other connections
between large deviation theory and statistical mechanics, in an effort to show that the mathematical
language of statistical mechanics is the language of large deviation theory. The first part of
the review presents the basics of large deviation theory and works out many of its classical applications
related to sums of random variables and Markov processes. The second part goes through many
problems and results of statistical mechanics, and shows how these can be formulated and derived
within the context of large deviation theory. The problems and results treated cover a wide range of
physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium
systems, as well as multifractals, disordered systems, and chaotic systems. This review also
covers many fundamental aspects of statistical mechanics, such as 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