I will present our first attempts to prove error estimates for finite element approximations of a parabolic stochastic partial differential equation: du + Au dt = dW where A is an elliptic operator and W is space-time white noise. Based on appropriate nonsmooth data error estimates for deterministic finite element problems, we obtain the error estimates in semidiscrete and fully discrete cases in both strong and weak norms. Our results are general and can be applied to several spatial variables.