Solutions of the linearized Boltzmann equation which may be found from finite moment equations are studied. A general solution to this problem is found and its properties discussed. A more restrictive class of such solutions, called generalized normal solutions, are then uncovered. These constitute a wide class of solutions in any desired number of moments. The governing equations are rendered determinate by exact expressions relating higher moments to the distinguished moments. In certain circumstances the initial data must be altered, and the resulting ``ersatz'' initial data used in connection with the equations. A case in point is the usual normal solution of the Hilbert-Chapman-Enskog theory. The ``ersatz'' initial data then renders the latter into an exact asymptotic theory. In addition higher moment systems are discussed in detail. It is shown that the generalized normal solutions are by no means a comprehensive class. Exact solutions of the Boltzmann equation which are also exact solutions of the Euler equations, the Navier-Stokes equations, the Burnett equations, the Thirteen moments equations (among others) are exhibited. Furthermore, these can satisfy the Boltzmann equation in an infinite variety of ways.