A central question in dynamics is whether the topology of a system
determines its geometry, whether the system is rigid. Under mild
topological conditions rigidity holds in many classical cases,
including: Kleinian groups, circle diffeomorphisms, unimodal interval
maps, critical circle maps, and circle maps with a break point. More
recent developments show that under similar topological conditions,
rigidity does not hold for slightly more general systems. We will
discuss the case of circle maps with a flat interval. The class of
maps with Fibonacci rotation numbers is a C^1 manifold which is
foliated with co dimension three rigidity classes. Finally, we
summarize the known non-rigidity phenomena in a conjecture which
describes how topological classes are organized into rigidity classes.