  In this essay, we first sketch the development of ideas on the extraordinary
dynamics of integrable classical nonlinear and quantum many body Hamiltonians.
In particular, we comment on the state of mathematical techniques available for
analyzing their thermodynamic and dynamic properties.
  Then, we discuss the unconventional finite temperature transport of
integrable systems using as example the classical Toda chain and the toy model
of a quantum particle interacting with a fermionic bath in one dimension; we
focus on the singular long time asymptotic of current-current correlations, we
introduce the notion of the Drude weight and we emphasize the role played by
conservation laws in establishing the ballistic character of transport in these
systems.
