  In this paper, we consider state and control path-dependent stochastic
zero-sum differential games, where the dynamics and the running cost include
both state and control paths of the players. Using the notion of
nonanticipative strategies, we define lower and upper value functionals, which
are functions of the initial state and control paths of the players. We prove
that the value functionals satisfy the dynamic programming principle. The
associated lower and upper Hamilton-Jacobi-Isaacs (HJI) equations from the
dynamic programming principle are state and control path-dependent nonlinear
second-order partial differential equations. We apply the functional It\^o
calculus to prove that the lower and upper value functionals are viscosity
solutions of (lower and upper) state and control path-dependent HJI equations,
where the notion of viscosity solutions is defined on a compact subset of an
$\kappa$-H\"older space introduced in \cite{Tang_DCD_2015}. Moreover, we show
that the Isaacs condition and the uniqueness of viscosity solutions imply the
existence of the game value. For the state path-dependent case, we prove the
uniqueness of classical solutions for the (state path-dependent) HJI equations.
