A finite group with a given set of generators may naturally be viewed as a metric space endowed with the word metric. Understanding its geometry is a basic question of geometric group theory. One may ask, for example, when this metric space can be embedded with low distortion into a normed space (in other words, "can the word metric be approximately described a norm"?) My aim in this talk is to explain a connection between this problem and the properties of random walks on groups. Even the simplest possible example where this connection is fully understood, the discrete cube, settled a long-standing problem of Enflo in the geometry of Banach spaces (joint work with Ivanisvili and Volberg). In other groups, however, this program gives rise to natural questions about the behavior of random walks that do not appear to have been studied in the probability literature. I aim to explain ongoing work with Mira Gordin on the symmetric group, and some challenges we encounter in understanding more general situations.