Given a vertex-weighted undirected graph G = (V, E,w) and a positive integer k, we consider the k-separator problem: it consists in finding a minimum-weight subset of vertices whose removal leads to a graph where the size of each connected component is less than or equal to k. We show that this problem can be solved in polynomial time for some graph classes: for cycles and trees by a dynamic programming approach and by using a peculiar graph transformation coupled with recent results from the literature for mK2-free, (G1,G2, G3, P6)-free, interval-filament, asteroidal triple-free, weakly chordal, interval and circular-arc graphs. Approximation algorithms are also presented.