This paper describes a number of novel discontinuous finite element approximations to the linear Boltzmann transport equation. Two distinct approaches are used: one based on a dual set of basis functions and the other based on an upwinded characteristic method. In both approaches the boundary conditions are incorporated in a natural manner. The objective of the work is to develop a method that can follow abruptly-changing or nearly-discontinuous field variables, maintain solution feasibility e.g. non-negative physical quantities and achieve sparse direct coupling of the nodal moments when spherical harmonic angular descriptions arc used. The representative examples indicate that the methods offer attractive discretization methods for first-order partial differential equations. Appropriate solution methods for the resulting discretizations are also discussed.