  The first law of black hole mechanics has been the main motivation for
investigating thermodynamic properties of black holes. The first version of
this law was proved in \cite{Bardeen:1973gs} by considering perturbations of an
asymptotically flat, stationary black hole spacetime to other stationary black
hole spacetimes. This result was then extended to fully general perturbations,
first in the context of Einstein-Maxwell theory in
\cite{Sudarsky:1992ty},\cite{Wald:1993ki}, and then in the context of a general
diffeomorphism invariant theory of gravity with an arbitrary number of matter
fields in \cite{Wald:1993nt},\cite{Iyer:1994ys}. Here a review of these two
generalizations of the first law is presented, with particular attention to
outlining the necessary formalisms and calculations in an explicit and thorough
way, understandable at a graduate level. The open problem of defining the
entropy for a dynamical black hole that satisfies a form of the second law of
black hole mechanics is briefly discussed.
