Several researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes (ie. filtrations). We find that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover, we explain how optimal transport theory can be used to obtain a compatible metric that is both tractable and highly relevant for a number of practical questions. (Based on joint works with D. Bartl, M. Beiglboek, M. Eder.)