  Differentiable conjugacies link dynamical systems that share properties such
as the stability multipliers of corresponding orbits. It provides a stronger
classification than topological conjugacy, which only requires qualitative
similarity. We describe some of the techniques and recent results that allow
differentiable conjugacies to be defined for standard bifurcations, and explain
how this leads to a new class of normal forms. Closed-form expressions for
differentiable conjugacies exist between some chaotic maps, and we describe
some of the constraints that make it possible to recognise when such
conjugacies arise. This paper focuses on the consequences of the existence of
differentiable conjugacies rather than the conjugacy classes themselves.
