The Galerkin formulation of the finite element method is applied in space and angle to the equivalent integral law, or weak form, of the first-order neutron transport equation. The existence of a unique solution to the resultant system of algebraic equations is demonstrated using the positivity of the transport operator. Numerical results are given for the one-dimensional plane geometry application, including comparisons with the one-dimensional discrete ordinates code ANISN. A problem with strong heterogeneities is considered, and the use of discontinuous angular and spatial finite elements is shown to result in a marked improvement in the results. The success of the discontinuous elements is examined and it is seen that the discontinuous angular elements effectively match the analytical discontinuity in the angular flux at µ = 0 for plane geometry. Also, the use of discontinuous spatial elements is shown to result in treating continuity of the angular flux at an interface as a natural interface condition in the direction of neutron travel.