In this paper, we present a theoretical framework for studying coherent acceptability indices (CAIs) in a dynamic setup. We study dynamic CAIs (DCAIs) and dynamic coherent risk measures (DCRMs), and we establish a duality between them. We derive a representation theorem for DCRMs in terms of a so‐called dynamically consistent sequence of sets of probability measures. Based on these results, we give a specific construction of DCAIs. We also provide examples of DCAIs, both abstract and also some that generalize selected classical financial measures of portfolio performance.