This paper provides a comprehensive review of integration algorithms for finite-dimensional mechanical systems based on discrete variational principles. The variational approach unifies 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. Specific schemes discussed include Verlet, SHAKE, RATTLE, Newmark, and symplectic partitioned Runge–Kutta methods. The paper covers discrete variational mechanics from both Lagrangian and Hamiltonian perspectives, and explores the correspondence between discrete and continuous mechanics. It also delves into forcing and constrained systems, time-dependent mechanics, and further topics such as discrete symmetry reduction and multisymplectic integrators for PDEs. The authors highlight the importance of geometric integration techniques and their potential for handling constraints in mechanical systems.This paper provides a comprehensive review of integration algorithms for finite-dimensional mechanical systems based on discrete variational principles. The variational approach unifies 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. Specific schemes discussed include Verlet, SHAKE, RATTLE, Newmark, and symplectic partitioned Runge–Kutta methods. The paper covers discrete variational mechanics from both Lagrangian and Hamiltonian perspectives, and explores the correspondence between discrete and continuous mechanics. It also delves into forcing and constrained systems, time-dependent mechanics, and further topics such as discrete symmetry reduction and multisymplectic integrators for PDEs. The authors highlight the importance of geometric integration techniques and their potential for handling constraints in mechanical systems.