  We study the Cauchy problem for fully nonlinear (stochastic) parabolic
partial differential equations. We provide both in deterministic and stochastic
case the existence of a maximal defined solution for the problem and we provide
suitable blow-up criterion. The key idea is the use of the paradifferential
operator calculus in order to reduce the fully nonlinear problem into an
abstract quasilinear (stochastic) parabolic equation. This allows us to use
some recent results on abstract quasilinear (stochastic) evolution equations in
Banach spaces. To do so, we analyse the properties of the paradifferential
operator, in light of known results on the boundedness of the
$\mathcal{H}^{\infty}$-calculus for pseudodifferential operator. Finally, we
extend the theory just developed to cover high order fully nonlinear parabolic
(S)PDEs.
