  It is well known that general relativity (GR) does not possess any
non-trivial local (in a precise standard sense) and diffeomorphism invariant
observables. We propose a generalized notion of local observables, which retain
the most important properties that follow from the standard definition of
locality, yet is flexible enough to admit a large class of diffeomorphism
invariant observables in GR. The generalization comes at a small price, that
the domain of definition of a generalized local observable may not cover the
entire phase space of GR and two such observables may have distinct domains.
However, the subset of metrics on which generalized local observables can be
defined is in a sense generic (its open interior is non-empty in the Whitney
strong topology). Moreover, generalized local gauge invariant observables are
sufficient to separate diffeomorphism orbits on this admissible subset of the
phase space. Connecting the construction with the notion of differential
invariants, gives a general scheme for defining generalized local gauge
invariant observables in arbitrary gauge theories, which happens to agree with
well-known results for Maxwell and Yang-Mills theories.
