Memory effects play a fundamental role in the study of the dynamics of open quantum systems. There exist two conceptually distinct notions of memory discussed for quantum channels in the literature. In quantum information theory quantum channels with memory are characterized by the existence of correlations between successive applications of the channel on a sequence of quantum systems. In open quantum systems theory memory effects arise dynamically during the time evolution of quantum systems and define non-Markovian dynamics. Here we relate and combine these two different concepts of memory. In particular, we study the interplay between correlations between multiple uses of quantum channels and non-Markovianity as nondivisibility of the t-parametrized family of channels defining the dynamical map.