For every infinity-category C, there is a homotopy n-category h(n)C and a canonical functor gamma(n) : C -> h(n)C. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Using homotopy n-categories, we introduce the notion of an n-derivator and study the main examples arising from infinity-categories. Following the work of Maltsiniotis and Garkusha, we define K-theory for infinity-derivators and prove that the canonical comparison map from the Waldhausen K-theory of C to the K-theory of the associated n-derivator D-C((n)) is (n + 1)-connected. We also prove that this comparison map identifies derivator K-theory of infinity-derivators in terms of a universal property. Moreover, using the canonical structure of higher weak pushouts in the homotopy n-category, we also define a K-theory space K(h(n)C, can) associated to h(n)C. We prove that the canonical comparison map from the Waldhausen K-theory of C to K(h(n)C, can) is n-connected.