  We prove a priori gradient bounds for classical solutions of the fully
nonlinear parabolic equation $$u_{t}=F(D^2u,D u,u,x,t).$$ The domain is the
torus {\mathbb{T}}^{d} of dimension $d\ge1$. Up to the price of technicalities,
our work can be extended to the case of bounded domains or the case of the
whole space ${\mathbb{R}}^d$. Several applications are given, including the
standard porous medium equation.
