The aim of these notes is to prove that any right exact functor between reasonable Waldhausen categories, that induces an equivalence at the level of homotopy categories, gives rise to a homotopy equivalence between the corresponding K-theory spectra. This generalizes a well known result of Thomason and Trobaugh. The ingredients, for this proof, are a generalization of the Waldhausen approximation theorem, and a simple combinatorial caracterization of derived equivalences. We also study simplicial localization of Waldhausen categories. We prove that a (homotopy) right exact functor induces an equivalence of homotopy categories if and only if it induces anequivalence of simplicial localizations. This allows to make the link with the K-theory of simplicial categories introduced by Toën and Vezzosi.