The possibility to manipulate the heat current represents a fascinating challenge  for the future, especially in view of the need of future society of providing a   sustainable supply of energy and due to the strong concerns about the environmental impact of the combustion of fossil fuels. However  along these lines there are severe difficulties both of theoretical and experimental nature. In particular  it turns out that manipulation of the heat current is much more difficult than the manipulation of the electric current. A deeper understanding of the properties of heat transport is needed. We discuss thermal rectification and thermoelectric energy conversion from the perspective of nonequilibrium statistical mechanics and dynamical systems theory. After preliminary considerations on the dynamical foundations of the phenomenological  Fourier law, we illustrate ways to control the phononic heat flow and design thermal diodes. Finally, we consider the coupled transport of heat and charge  and discuss several general mechanisms for optimizing the figure of merit of  thermoelectric efficiency.