  In this paper we are concerned with the maximum principle for quasi-linear
backward stochastic partial differential equations (BSPDEs for short) of
parabolic type. We first prove the existence and uniqueness of the weak
solution to quasi-linear BSPDE with the null Dirichlet condition on the lateral
boundary. Then using the De Giorgi iteration scheme, we establish the maximum
estimates and the global maximum principle for quasi-linear BSPDEs. To study
the local regularity of weak solutions, we also prove a local maximum principle
for the backward stochastic parabolic De Giorgi class.
