The aim of the work presented in this thesis is the construction and the study of numerical integrators in time to solve stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs). More precisely, we are interested in the convergence of the methods, in the weak sense and for sampling the invariant measure in the case of ergodic dynamics, introducing a new algebraic framework of trees for the computation of order conditions. We focus on the preservation of geometric properties by the numerical integrators, in particular invariants and constraints, as well as on the robustness of the methods in the case of multiscale problems.