A systematicnonperturbaiivederivation is presented of a whole hierarchy of closed systems of moment equations corresponding to any classical kinetic theory that describes a gas of identical particles under the influence of an external potential. The closure has two steps. The first ensures that every member of the hierarchy is hyperbolic, has an entropy, and formally recovers the Euler limit The second modifies the collision operator so that members of the hierarchy beyond the second also recover the correct Navier-Stokes behavior. This is achieved through a generalization of the BGK collision operator.