The method of half-range polynomials is applied to neutron transport theory. The specific applicability of this method to problems having discontinuities in the nuclear parameters at the boundaries or interfaces is discussed. Half-range polynomial expansions are used to obtain solutions for both finite and semi-infinite slabs, which consist of isotropically scattering media. The results indicate that the half-range approximations compare favorably with higher approximations obtained from the full-range spherical harmonic or several discrete ordinate methods. In particular, the poor convergence, found in the full-range methods in regions close to the discontinuity, is not present in the half-range method. The latter method is used to obtain a pair of second-order coupled differential equations, as in diffusion theory.