Localization errors in solving stochastic partial differential equations in the whole space Cauchy problems with SPDEs on the whole space are localized to Cauchy problems on a ball of radius R. This localization reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localization and the space and time discretization, and thus is fully implementable. 86 307 2373 - 2397 2373 - 2397 American Mathematical Society