The idea that a noncommutative algebra which is a quantisation of a commutative Poisson algebra should have structure and representation theory which reflects the Poisson structure of the underlying commutative algebra (or variety) has a long history - for example, it lies at the heart of Kirillov's orbit method. I will review these concepts, describe a little of the history, and explain some recent results and some conjectures in this area, with specific reference to some or all of the following: enveloping algebras of Lie algebras; quantised function algebras of semisimple algebraic groups; and symplectic reflection algebras.