The talk presents an overview of selected results regarding recent achievements in the field of structure-preserving numerical integration of deterministic and stochastic differential equations. The proposed methodology leads to problem-oriented numerical schemes, able to accurately and efficiently reproduce typical properties of the continuous problem along the discrete dynamics. These features include, for instance, preservation of invariants for Hamiltonian problems, retaining the long term dynamics of stochastic oscillators, achieving exponential mean-square contractivity for nonlinear stochastic differential equations, retaining periodic wavefronts in reaction-diffusion problems, and so on. The presentation aims to show the benefits of merging a-priori achievable informations on the problem into the numerical scheme, with a significant gain in accuracy and efficiency with respect to the classical case of general purpose numerical methods. The presented results deals with the joint research described in a selection of recent papers in collaboration with E. Buckwar (Johannes Kepler University of Linz), J.C. Butcher (University of Auckland), L. Dieci (Georgia Institute of Technology), E. Hairer (University of Geneva), B. Paternoster (University of Salerno).