A geometric integrator is a numerical method that preserves the geometric properties of the flow of a differential equation. The rise of geometric integrators stems from the geometrization of mechanics and the development of numerical analysis. I will explain here their relevance within the framework of Poisson geometry, which accounts for a broad class of conservative mechanical systems. Throughout the presentation, examples will illustrate their qualitative properties, such as symmetry preservation or behavior in the vicinity of a singularity.