  We consider the kinetic theory of the quantum and classical Toda lattice
models. A kinetic equation of Bethe-Boltzmann type is derived for the
distribution function of conserved quasiparticles. Near the classical limit, we
show that the kinetic theory depends smoothly on Planck's constant, and
explicitly characterise the leading quantum corrections to classical behaviour.
The classical kinetic theory is compared with direct numerical simulations and
shows excellent agreement. Finally, we connect the Bethe-Boltzmann approach
with the classical inverse scattering method, by identifying conserved
quasiparticles with the spectrum of the Lax matrix.
