  We introduce an $hp$-version discontinuous Galerkin finite element method
(DGFEM) for the linear Boltzmann transport problem. A key feature of this new
method is that, while offering arbitrary order convergence rates, it may be
implemented in an almost identical form to standard multigroup discrete
ordinates methods, meaning that solutions can be computed efficiently with high
accuracy and in parallel within existing software. This method provides a
unified discretisation of the space, angle, and energy domains of the
underlying integro-differential equation and naturally incorporates both local
mesh and local polynomial degree variation within each of these computational
domains. Moreover, general polytopic elements can be handled by the method,
enabling efficient discretisations of problems posed on complicated spatial
geometries. We study the stability and $hp$-version a priori error analysis of
the proposed method, by deriving suitable $hp$-approximation estimates together
with a novel inf-sup bound. Numerical experiments highlighting the performance
of the method for both polyenergetic and monoenergetic problems are presented.
