Stochastic partial differential equations (SPDEs) are used more and more often to model real-world phenomena. Currently, statistical methodology for these equations driven by space-time white noise is developing rapidly. Based on the classical spectral method for parametric drift estimation, we shall exhibit fundamental differences with the case of stochastic ordinary differential equations. This method, however, is restricted to simple parametric situations and we discuss the local estimation method in detail, which allows to estimate varying coefficients in the differential operator of a parabolic SPDE nonparametrically with optimal rates. This approach is extended to observations under measurement errors (‘static noise’), showing a fundamentally different impact of dynamic and static noise levels. Finally, we present an abstract minimax lower bound framework for stochastic evolution equations generated by normal operators in Hilbert space and obtain a rich picture of complexity for different SPDE estimation objectives. Some illustrations with cell motility experiments in biophysics are provided.