  We review the quantum version of the linear Boltzmann equation, which
describes in a non-perturbative fashion, by means of scattering theory, how the
quantum motion of a single test particle is affected by collisions with an
ideal background gas. A heuristic derivation of this Lindblad master equation
is presented, based on the requirement of translation-covariance and on the
relation to the classical linear Boltzmann equation. After analyzing its
general symmetry properties and the associated relaxation dynamics, we discuss
a quantum Monte Carlo method for its numerical solution. We then review
important limiting forms of the quantum linear Boltzmann equation, such as the
case of quantum Brownian motion and pure collisional decoherence, as well as
the application to matter wave optics. Finally, we point to the incorporation
of quantum degeneracies and self-interactions in the gas by relating the
equation to the dynamic structure factor of the ambient medium, and we provide
an extension of the equation to include internal degrees of freedom.
