This talk aims to present a survey of some recent advances on structure-preserving numerical methods for stochastic differential equations, according to the basic principles of geometric numerical integration. Namely, we consider the following two tracks: - track 1: geometric numerical integration of stochastic Hamiltonian problems. For these problems, if the noise is driven in Ito sense, the expected Hamiltonian function shows a linear drift in time; in the Stratonovich case, the Hamiltonian is preserved along each path. In both cases, the talk is focused on analyzing the ability of selected numerical methods in preserving the aforementioned behaviors. A long term investigation via backward error analysis is also presented; - track 2: structure-preserving numerics of stochastic PDEs. In this case, the attention is focused on the stochastic Korteweg-de Vries equation, characterized by certain invariance laws for the exact dynamics. The talk focuses on their long-term conservation along the numerical dynamics provided by stochastic theta-methods for the time integration of the spatially discretized system. For both tracks, numerical evidence on sustainability problems, supporting the theoretical investigation, will be provided. The talk falls within the activities of the PRIN 2022 project 20229P2HEA “Stochastic numerical modelling for sustainable innovation”, CUP: E53D23017940001, granted by the Italian Ministry of University and Research within the framework of the Call relating to the scrolling of the final rankings of the PRIN 2022 call.