The idea is to construct numerical integrator methods for Hamiltonian type of ODE’s which are defined in an ambient Poisson geometry. The goal is to approximate the exact dynamical solutions of this ODE while, at the same time, preserve the Poisson structure to a certain controlled degree. This is a non-trivial and long-range generalization of the notion of symplectic method in which the Poisson geometry is non-degenerate, thus, symplectic. We first outline a first approach to such methods which uses the geometry of so-called approximate symplectic realizations based on recent joint work with D. Martín de Diego and M. Vaquero. Finally, we describe a second approach based on theoretical results coming from Lie-theoretic aspects and which use an underlying groupoid multiplication, based on work in progress with D. Iglesias and J.C. Marrero.