In order to accelerate computations, improve long time accuracy of numerical simulations, and sample statistics distribution by dynamics, we develop multiscale geometric integrators. Most of the talk will be focused on the description of FLow AVeraging integratORs (FLAVORs), which apply to general multiscale stiff ODEs, SDEs, and PDEs. These integrators employ coarse integration steps that do not resolve the fast timescale in the dynamics; nevertheless, they capture the correct effective contribution of the fast dynamics. Distinct from existing approaches, an identification of the underlying slow variables (or process) is not required, and intrinsic geometric structures (e.g. symplecticity, conservation laws) can be preserved by the multiscale simulation.