Abstract. The existence of maximal and minimal solutions for initialboundary value problems and the Cauchy initial value problem associated with Lu -f(x, t, u, Vu) where L is a second order uniformly parabolic differential operator is obtained by constructing maximal and minimal solutions from all possible lower and all possible upper solutions, respectively. This approach allows / to be highly nonlinear, i.e., / locally Holder continuous with almost quadratic growth in |V«|.