This overview focuses on splitting methods, a class of numerical integrators designed for differential equations that can be decomposed into simpler subproblems. These methods are closely related to composition methods, which combine one or more low-order schemes to achieve higher-order approximations. The authors analyze the order conditions necessary for these methods to achieve a given order and discuss their qualitative properties, particularly in the context of geometric numerical integration and the treatment of highly oscillatory problems. The survey also covers the application of splitting methods to partial differential equations, with a focus on parabolic equations. It includes a comprehensive list of methods of different orders and their testing on simple examples. Additionally, the authors provide applications of splitting methods in various fields, ranging from celestial mechanics to statistics. The content is structured into sections that cover the introduction to splitting methods, their implementation in flows and differential operators, adjoint and conjugate methods, and their application to specific problems such as the mathematical pendulum, the gravitational N-body problem, and the time-dependent Schrödinger equation. The overview emphasizes the importance of splitting methods in preserving structural properties of the exact solution, making them superior to standard integrators in certain contexts, especially for long-time integrations.This overview focuses on splitting methods, a class of numerical integrators designed for differential equations that can be decomposed into simpler subproblems. These methods are closely related to composition methods, which combine one or more low-order schemes to achieve higher-order approximations. The authors analyze the order conditions necessary for these methods to achieve a given order and discuss their qualitative properties, particularly in the context of geometric numerical integration and the treatment of highly oscillatory problems. The survey also covers the application of splitting methods to partial differential equations, with a focus on parabolic equations. It includes a comprehensive list of methods of different orders and their testing on simple examples. Additionally, the authors provide applications of splitting methods in various fields, ranging from celestial mechanics to statistics. The content is structured into sections that cover the introduction to splitting methods, their implementation in flows and differential operators, adjoint and conjugate methods, and their application to specific problems such as the mathematical pendulum, the gravitational N-body problem, and the time-dependent Schrödinger equation. The overview emphasizes the importance of splitting methods in preserving structural properties of the exact solution, making them superior to standard integrators in certain contexts, especially for long-time integrations.