This paper presents a review of integration algorithms for finite-dimensional mechanical systems based on discrete variational principles. The variational approach provides a unified treatment of many symplectic schemes, including higher-order methods, and naturally incorporates the discrete Noether theorem. It also allows for the inclusion of forces, dissipation, and constraints in a natural way. Examples of variational integrators include the Verlet, SHAKE, RATTLE, Newmark, and symplectic partitioned Runge–Kutta schemes. The paper is divided into five parts. Part 1 introduces discrete variational mechanics from both Lagrangian and Hamiltonian viewpoints. It discusses the correspondence between discrete and continuous mechanics and the Hamilton–Jacobi theory. Part 2 focuses on variational integrators, covering error analysis, the adjoint of a method, composition methods, and examples of variational integrators. Part 3 addresses forcing and constraints, including the treatment of forced and constrained systems. Part 4 discusses time-dependent mechanics, extending the variational approach to time-dependent systems. Part 5 covers further topics, including discrete symmetry reduction and multisymplectic integrators for PDEs. The paper emphasizes the importance of the variational approach in deriving symplectic integrators that preserve key geometric properties of mechanical systems, such as energy and momentum conservation. It also highlights the role of discrete variational mechanics in handling constraints and forces, and its connection to other areas of mechanics, such as continuum mechanics and systems with forcing and constraints. The paper provides a detailed derivation of the discrete Euler–Lagrange equations and shows how they lead to symplectic integrators that preserve the structure of the phase space. It also discusses the discrete Noether theorem, which ensures the conservation of momentum maps under symmetry actions. The paper concludes with a discussion of the broader implications of the variational approach for the numerical integration of mechanical systems.This paper presents a review of integration algorithms for finite-dimensional mechanical systems based on discrete variational principles. The variational approach provides a unified treatment of many symplectic schemes, including higher-order methods, and naturally incorporates the discrete Noether theorem. It also allows for the inclusion of forces, dissipation, and constraints in a natural way. Examples of variational integrators include the Verlet, SHAKE, RATTLE, Newmark, and symplectic partitioned Runge–Kutta schemes. The paper is divided into five parts. Part 1 introduces discrete variational mechanics from both Lagrangian and Hamiltonian viewpoints. It discusses the correspondence between discrete and continuous mechanics and the Hamilton–Jacobi theory. Part 2 focuses on variational integrators, covering error analysis, the adjoint of a method, composition methods, and examples of variational integrators. Part 3 addresses forcing and constraints, including the treatment of forced and constrained systems. Part 4 discusses time-dependent mechanics, extending the variational approach to time-dependent systems. Part 5 covers further topics, including discrete symmetry reduction and multisymplectic integrators for PDEs. The paper emphasizes the importance of the variational approach in deriving symplectic integrators that preserve key geometric properties of mechanical systems, such as energy and momentum conservation. It also highlights the role of discrete variational mechanics in handling constraints and forces, and its connection to other areas of mechanics, such as continuum mechanics and systems with forcing and constraints. The paper provides a detailed derivation of the discrete Euler–Lagrange equations and shows how they lead to symplectic integrators that preserve the structure of the phase space. It also discusses the discrete Noether theorem, which ensures the conservation of momentum maps under symmetry actions. The paper concludes with a discussion of the broader implications of the variational approach for the numerical integration of mechanical systems.