  We study fully nonlinear second-order (forward) stochastic partial
differential equations (SPDEs). They can also be viewed as forward
path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a
unified framework. We develop first a local theory of classical solutions and
define then viscosity solutions through smooth test functions. Our notion of
viscosity solutions is equivalent to the alternative one using semi-jets. Next,
we prove basic properties such as consistency, stability, and a partial
comparison principle in the general setting. When the diffusion coefficient is
semi-linear (but the drift can be fully nonlinear), we establish a complete
theory, including global existence and comparison principle. Our methodology
relies heavily on the method of characteristics.
