  In this paper, a class of non-Markovian forward-backward doubly stochastic
systems is studied. By using the technique of functional It\^o (or
path-dependent) calculus, the relationship between the systems and related
path-dependent quasi-linear stochastic partial differential equations (SPDEs in
short) is established, and the well-known nonlinear stochastic Feynman-Kac
formula of Pardoux and Peng [Backward doubly stochastic differential equations
and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98 (1994), pp.
209--227] is developed to the non-Markovian situation. Moreover, we obtain the
differentiability of the solution to the forward-backward doubly stochastic
systems and some properties of solutions to the path-dependent SPDEs.
