In this talk we will come at this question from two different angles: first, from the viewpoint of model theory, a subject in which for nearly half a century the notion of stability has played a central role in describing tame behaviour\DSEMIC secondly, from the perspective of combinatorics, where so-called regularity decompositions have enjoyed a similar level of prominence in a range of finitary settings, with remarkable applications including to patterns in the primes.

In recent years, these two fundamental notions have been shown to interact in interesting ways. In particular, it has been shown that mathematical objects that are stable in the model-theoretic sense admit particularly well-behaved regularity decompositions. In this talk we will explore this fruitful interplay in the context of both finite graphs and subsets of abelian groups.