  Many important features of a field theory, {\it e.g.}, conserved currents,
symplectic structures, energy-momentum tensors, {\it etc.}, arise as tensors
locally constructed from the fields and their derivatives. Such tensors are
naturally defined as geometric objects on the jet space of solutions to the
field equations. Modern results from the calculus on jet bundles can be
combined with a powerful spinor parametrization of the jet space of Einstein
metrics to unravel basic features of the Einstein equations. These techniques
have been applied to computation of generalized symmetries and ``characteristic
cohomology'' of the Einstein equations, and lead to results such as a proof of
non-existence of ``local observables'' for vacuum spacetimes and a uniqueness
theorem for the gravitational symplectic structure.
