Two stochastic processes can have similar laws but yield a vastly different outcome in applications such as optimal stopping or stochastic programming. The reason is that the usual weak topology does not account for the different available information in time that is stored in the filtration of the underlying process. To address the resulting discontinuities, Aldous introduced the extended weak topology, and subsequently, Hoover and Keisler showed that both, weak topology as well as extended weak topology, are just the first two topologies in a sequence of the so-called adapted weak topologies that get increasingly finer. In this short survey, we give a brief introduction to the recent advances of the applications of signature theory, in particular the higher-rank (expected) signatures, in adapted weak topologies and related fields, and highlight theoretical and computational properties.