This Summer School treats numerical methods that preserve geometric properties of the flow of a differential equation: symplectic integrators for Hamiltonian systems, symmetric integrators for reversible systems, methods preserving first integrals and numerical methods on manifolds. The main issue is long-time integration, which can be understood with the help of backward error analysis for ordinary differential equations, and with modulated Fourier expansions for partial differential equations (such as the wave equation and the Schrödinger equation). Much of the material is taken from the monograph "Geometric Numerical Integration" (2nd edition, 2006), but also more recent developments will be presented.