The mathematical foundations of transport properties are analyzed in detail in several Hamiltonian dynamical models. Deterministic diffusion is studied in the multibaker map and the Lorentz gases where a point particle moves in a two-dimensional lattice of hard disks or Yukawa potentials. In these chaotic models, the diffusive modes are constructed as the eigenmodes of the Liouvillian dynamics associated with Pollicott–Ruelle resonances. These eigenmodes are distributions with a fractal cumulative function. As a consequence of this fractal character, the entropy production calculated by coarse graining has the expression expected for diffusion in nonequilibrium thermodynamics. Furthermore, Fourier’s law for heat conduction is shown to hold in many-particle billiard models, where heat conductivity can be evaluated with very high accuracy at a conductor-insulator transition. Finally, mechanothermal coupling is illustrated with models for motors propelled by a temperature difference.