In the 1930s, John von Neumann developed a continuous-dimensional analogue of finite-dimensional projective geometry. Inspired by conversations with Garrett Birkhoff as well as his collaboration with Francis Murray on rings of operators, von Neumann introduced and studied the notion of a continuous geometry, which is a complete complemented modular lattice possessing a certain continuity property. Among other remarkable results, von Neumann proved that every continuous geometry of order at least four can be coordinatized by some (up to isomorphism unique) ring, and that the continuous rings (i.e., rings corresponding to continuous geometries via this coordinatization theorem) are precisely those irreducible, regular rings which admit a complete rank function. The necessarily unique rank function of a continuous ring gives rise to a compatible metric and thus furnishes the ring with a natural topology. Unit groups of such continuous rings, equipped with the relative topology, constitute an interesting family of topological groups with many peculiar dynamical properties. The talk will provide an introduction to von Neumann’s continuous geometry and discuss some of the latest advances concerning topological dynamics of unit groups of continuous rings.