  We discuss the problem of heat conduction in classical and quantum low
dimensional systems from a microscopic point of view. At the classical level we
provide convincing numerical evidence for the validity of Fourier law of heat
conduction in linear mixing systems, i.e. in systems without exponential
instability. At the quantum level, where motion is characterized by the lack of
exponential dynamical instability, we show that the validity of Fourier law is
in direct relation with the onset of quantum chaos. We then study the
phenomenon of thermal rectification and briefly discuss the different types of
microscopic mechanisms that lead to the rectification of heat flow. The control
of heat conduction by nonlinearity opens the possibility to propose new devices
such as a thermal rectifier.
