\noindent many interesting examples of dynamical systems involve several well separated time scales. in many applications, for example in molecular dynamics simulations, one is only interested in the slow aspects of dynamics, or on the long-time behavior of the solutions. however, when the different scales are coupled, small or fast perturbations can build up to an observable effect that cannot be neglected. \\ \noindent in this talk i will discuss several types of models and address some of the analytic and computational difficulties common to many systems evolving on multiple time scales. we give a complete characterization of the slow aspects of the dynamics and devise efficient computational algorithms that take advantage of the scale separation. it is shown that the computational cost is practically independent of the spectral gap. among the systems studied are highly oscillatory odes and a benchmark model of elastic spheres with disparate masses.