This paper provides an overview of splitting methods, a class of numerical integrators for differential equations that decompose the problem into simpler subproblems. These methods are closely related to composition methods, which combine low-order schemes to achieve higher-order approximations. The paper analyzes the order conditions required for these methods and discusses their qualitative properties, including geometric numerical integration and the treatment of highly oscillatory problems. It also covers splitting methods for partial differential equations, with a focus on parabolic equations. An extensive list of methods of different orders is tested on simple examples, and applications of splitting methods in various areas, such as celestial mechanics and statistics, are provided. The paper begins with an introduction to Lie–Trotter and Strang methods, which are fundamental examples of splitting methods. These methods approximate the solution of differential equations by decomposing the problem into simpler subproblems that can be solved more easily. The Lie–Trotter method is a first-order method, while the Strang method is a second-order method. The paper discusses the properties of these methods, including their symplectic nature for Hamiltonian systems and their ability to preserve certain geometric properties of the solution. The paper then explores the concept of adjoint methods and conjugate methods, which are important for ensuring the accuracy and stability of numerical solutions. It discusses the mathematical pendulum as an example of a nonlinear Hamiltonian system and shows how splitting methods can be applied to preserve the energy of the system. The paper also presents the gravitational N-body problem as another example, where splitting methods are used to simulate the motion of celestial bodies. The paper further discusses the time-dependent Schrödinger equation, a fundamental equation in quantum mechanics, and shows how splitting methods can be applied to approximate the solution of this equation. The paper highlights the importance of splitting methods in preserving the unitary nature of the solution and their ability to handle highly oscillatory problems. Overall, the paper provides a comprehensive overview of splitting methods, their theoretical foundations, and their applications in various areas of science and engineering. It emphasizes the importance of these methods in preserving the qualitative properties of the solutions to differential equations and their effectiveness in handling complex problems.This paper provides an overview of splitting methods, a class of numerical integrators for differential equations that decompose the problem into simpler subproblems. These methods are closely related to composition methods, which combine low-order schemes to achieve higher-order approximations. The paper analyzes the order conditions required for these methods and discusses their qualitative properties, including geometric numerical integration and the treatment of highly oscillatory problems. It also covers splitting methods for partial differential equations, with a focus on parabolic equations. An extensive list of methods of different orders is tested on simple examples, and applications of splitting methods in various areas, such as celestial mechanics and statistics, are provided. The paper begins with an introduction to Lie–Trotter and Strang methods, which are fundamental examples of splitting methods. These methods approximate the solution of differential equations by decomposing the problem into simpler subproblems that can be solved more easily. The Lie–Trotter method is a first-order method, while the Strang method is a second-order method. The paper discusses the properties of these methods, including their symplectic nature for Hamiltonian systems and their ability to preserve certain geometric properties of the solution. The paper then explores the concept of adjoint methods and conjugate methods, which are important for ensuring the accuracy and stability of numerical solutions. It discusses the mathematical pendulum as an example of a nonlinear Hamiltonian system and shows how splitting methods can be applied to preserve the energy of the system. The paper also presents the gravitational N-body problem as another example, where splitting methods are used to simulate the motion of celestial bodies. The paper further discusses the time-dependent Schrödinger equation, a fundamental equation in quantum mechanics, and shows how splitting methods can be applied to approximate the solution of this equation. The paper highlights the importance of splitting methods in preserving the unitary nature of the solution and their ability to handle highly oscillatory problems. Overall, the paper provides a comprehensive overview of splitting methods, their theoretical foundations, and their applications in various areas of science and engineering. It emphasizes the importance of these methods in preserving the qualitative properties of the solutions to differential equations and their effectiveness in handling complex problems.