Abstract. For nonautonomous linear difference equations in a Banach space and admitting a very general type of dichotomy, we show: i) the existence of global invariant manifolds for small Lipschitz perturbations of the linear equation; ii) the existence of local invariant manifolds for locally Lipschitz perturbations of the linear equation; iii) the persistence of the dichotomic behavior under small linear perturbations exactly with the same growth rates. In the particular case of (μ,ν)-dichotomies, nonuniform exponential dichotomies and nonuniform polynomial dichotomies our results are new or improve previous known results. This talk is based on joint work with C.M. Silva.