  We develop the kinetic theory of classical and quantum particles (fermions
and bosons) in gravitational interaction. The kinetic theory of quantum
particles may have applications in the context of dark matter. For simplicity,
we consider an infinite and spatially homogeneous system (or make a local
approximation) and neglect collective effects. This leads to the quantum Landau
equation derived heuristically in [Chavanis, Physica A 332, 89 (2004)]. We
establish its main properties: conservation laws, $H$-theorem, equilibrium
state, relaxation time, quantum diffusion and friction coefficients, quantum
Rosenbluth potentials, self-consistent evolution, (thermal) bath approximation,
quantum Fokker-Planck equation, quantum King model... For bosonic particles,
the Landau equation can describe the process of Bose-Einstein condensation. We
discuss the relation of our study with the works of [Levkov et al., Phys. Rev.
Lett. 121, 151301 (2018); Bar-Or et al., Astrophys. J. 871, 28 (2019)] on fuzzy
dark matter halos and the formation of Bose stars and solitons.
