  It is well-known that principal bundles and associated bundles underlie the
geometric structure of classical gauge field theories. In this paper, we
explore the reformulation of gauge theories in terms of Lie algebroids and
their associated bundles. This turns out to be a simple but elegant change,
mathematically involving a quotient that removes spurious structure. The payoff
is that the entire geometric structure involves only vector bundles over
space-time, and we emphasize that familiar concepts such as BRST are built into
the geometry, rather than appearing as adjunct structure. Thus the formulation
of gauge theories in terms of Lie algebroids provides a fully off-shell account
of the BRST complex. We expect that this formulation will have appealing
impacts on the geometric understanding of quantization and anomalies, as well
as entanglement in gauge theories. The formalism covers all gauge theories, and
we discuss Yang-Mills theories with matter as well as gravitational theories
explicitly.
