  The main objective of this paper and the accompanying one \cite{ETZ2} is to
provide a notion of viscosity solutions for fully nonlinear parabolic
path-dependent PDEs. Our definition extends our previous work \cite{EKTZ},
focused on the semilinear case, and is crucially based on the nonlinear optimal
stopping problem analyzed in \cite{ETZ0}. We prove that our notion of viscosity
solutions is consistent with the corresponding notion of classical solutions,
and satisfies a stability property and a partial comparison result. The latter
is a key step for the wellposedness results established in \cite{ETZ2}. We also
show that the value processes of path-dependent stochastic control problems are
viscosity solutions of the corresponding path dependent dynamic programming
equation.
