The linear, space-inhomogeneous transport equation for a distribution function (describing, for instance, a neutron distribution) in a bounded body with general boundary conditions is considered. Results on weak convergence to equilibrium, when t→ ∞, are given for general initial data, first in the cutoff case and then for infinite-range collision forces. To handle these problems, general H theorems (concerning monotonicity in time of convex entropy functionals) are presented. Furthermore, general results on collision invariants, i.e., on functions satisfying detailed balance relations in a binary collision, are given.