This review presents the formalism and applications of non-linear perturbation theory (PT) to understand the large-scale structure of the Universe. It discusses the dynamics of gravitational instability, from linear to non-linear regimes, including Eulerian and Lagrangian PT, non-linear approximations, and numerical simulation techniques. The review covers statistical tools used in cosmology, such as correlation functions, probability distribution functions, cumulants, and generating functions. It then reviews the use of PT to make quantitative predictions about these statistics based on initial conditions, including effects of non-Gaussianity. Results are illustrated by comparisons with numerical simulations. The review also discusses applications to observations, including estimators for galaxy catalog statistics, galaxy bias, redshift distortions, projection effects, and weak gravitational lensing. It concludes with a discussion of the current observational status and future prospects for galaxy surveys. The review is structured to allow independent reading of chapters, with a focus on the density field and its statistical description. It includes detailed discussions of the equations of motion, the Vlasov equation, Eulerian and Lagrangian dynamics, and the evolution of density and velocity fields. The review also covers the application of PT to higher-order statistics, the role of non-linear growth factors, and the connection between PT and observational data. It concludes with a summary of key results and future directions in the field.This review presents the formalism and applications of non-linear perturbation theory (PT) to understand the large-scale structure of the Universe. It discusses the dynamics of gravitational instability, from linear to non-linear regimes, including Eulerian and Lagrangian PT, non-linear approximations, and numerical simulation techniques. The review covers statistical tools used in cosmology, such as correlation functions, probability distribution functions, cumulants, and generating functions. It then reviews the use of PT to make quantitative predictions about these statistics based on initial conditions, including effects of non-Gaussianity. Results are illustrated by comparisons with numerical simulations. The review also discusses applications to observations, including estimators for galaxy catalog statistics, galaxy bias, redshift distortions, projection effects, and weak gravitational lensing. It concludes with a discussion of the current observational status and future prospects for galaxy surveys. The review is structured to allow independent reading of chapters, with a focus on the density field and its statistical description. It includes detailed discussions of the equations of motion, the Vlasov equation, Eulerian and Lagrangian dynamics, and the evolution of density and velocity fields. The review also covers the application of PT to higher-order statistics, the role of non-linear growth factors, and the connection between PT and observational data. It concludes with a summary of key results and future directions in the field.