Abstract: We discuss several counterexamples to smooth rigidity conjectures which state that under some quantitative assumption on non-existence of periodic orbits, topological conjugacy implies $C^1$ (or even $C^\infty$) conjugacy. We construct examples of non-rigid diffeomorphisms on the $2$-torus, which satisfy such quantitative assumptions. We also construct examples of flows and maps which are topologically conjugate, but not $C^1$ conjugate. These latter examples are based on results on solutions of the cohomological equation and suggest that the structure of the space of invariant distributions has to be taken into account in rigidity questions. The discussion suggests that Diophantine skew-shifts on the $2$-dimensional torus may be smoothly rigid. This is joint work with Adam Kanigowski.