We consider stochastic partial differential equations of divergence and nondivergence forms with nonsmooth coefficients. We show that there exists a constant Po>2 such that if p ∈ [2,po), then there exists a unique solution in some stochastic Banach space   As a consequence, we obtain the Holder continuity of the solution. The improvement of the integrability from 2 to p allows us to make use of some embedding theorem for  -spaces. And since we do not require the continuity of the coefficients, our result can be applied in the study of certain nonlinear SPDEs