This paper reviews the formalism and applications of non-linear perturbation theory (PT) to understanding the large-scale structure of the Universe. It begins by discussing the dynamics of gravitational instability, from the linear to the non-linear regime, including Eulerian and Lagrangian PT, non-linear approximations, and numerical simulation techniques. The paper then covers basic statistical tools used in cosmology to describe cosmic fields, such as correlation functions, probability distribution functions, cumulants, and generating functions. It reviews how PT can be used to make quantitative predictions about these statistics based on initial conditions, including the effects of possible non-Gaussianity of primordial fields. The results are illustrated by detailed comparisons of PT predictions with numerical simulations. The paper also addresses applications to observations, including practical estimators of statistics in galaxy catalogs, cosmic bias, redshift distortions, projection effects, and weak gravitational lensing. Finally, it discusses the current observational situation regarding galaxy clustering and the potential of future galaxy surveys.This paper reviews the formalism and applications of non-linear perturbation theory (PT) to understanding the large-scale structure of the Universe. It begins by discussing the dynamics of gravitational instability, from the linear to the non-linear regime, including Eulerian and Lagrangian PT, non-linear approximations, and numerical simulation techniques. The paper then covers basic statistical tools used in cosmology to describe cosmic fields, such as correlation functions, probability distribution functions, cumulants, and generating functions. It reviews how PT can be used to make quantitative predictions about these statistics based on initial conditions, including the effects of possible non-Gaussianity of primordial fields. The results are illustrated by detailed comparisons of PT predictions with numerical simulations. The paper also addresses applications to observations, including practical estimators of statistics in galaxy catalogs, cosmic bias, redshift distortions, projection effects, and weak gravitational lensing. Finally, it discusses the current observational situation regarding galaxy clustering and the potential of future galaxy surveys.