  We consider a general, classical theory of gravity in $n$ dimensions, arising
from a diffeomorphism invariant Lagrangian. In any such theory, to each vector
field, $\xi^a$, on spacetime one can associate a local symmetry and, hence, a
Noether current $(n-1)$-form, ${\bf j}$, and (for solutions to the field
equations) a Noether charge $(n-2)$-form, ${\bf Q}$. Assuming only that the
theory admits stationary black hole solutions with a bifurcate Killing horizon,
and that the canonical mass and angular momentum of solutions are well defined
at infinity, we show that the first law of black hole mechanics always holds
for perturbations to nearby stationary black hole solutions. The quantity
playing the role of black hole entropy in this formula is simply $2 \pi$ times
the integral over $\Sigma$ of the Noether charge $(n-2)$-form associated with
the horizon Killing field, normalized so as to have unit surface gravity.
Furthermore, we show that this black hole entropy always is given by a local
geometrical expression on the horizon of the black hole. We thereby obtain a
natural candidate for the entropy of a dynamical black hole in a general theory
of gravity. Our results show that the validity of the ``second law" of black
hole mechanics in dynamical evolution from an initially stationary black hole
to a final stationary state is equivalent to the positivity of a total Noether
flux, and thus may be intimately related to the positive energy properties of
the theory. The relationship between the derivation of our formula for black
hole entropy and the derivation via ``Euclidean methods" also is explained.
