We present a method of solution of Boltzmann's equation that is valid for conditions where the species velocity distribution function departs significantly from a Maxwellian. This method should be particularly useful for problems in aeronomy and space physics where there are large or rapidly varying forces acting on a neutral or ionized gas or for problems dealing with highly rarefied gases where the mean free path for collisions is comparable to or greater than the characteristic scale length of the problem. The method of solution, which was first introduced by Mintzer, corresponds to expanding the species distribution function in a generalized orthogonal polynomial series with an arbitrary weight factor. The specific form of the weight factor depends on the details of the problem. The method is illustrated by calculating auroral ion velocity distributions for a relaxation collision model, and the results are compared with the velocity distributions obtained from an exact solution to Boltzmann's equation.