This paper proposes a model for, and investigates the consequences of, strong spatial dependence in economic variables. Our approach and findings echo those of the corresponding unit root time series literature: We suggest a model for spatial I(1) processes, and establish a functional central limit theorem that justifies a large sample Gaussian process approximation for such processes. We further generalize the I(1) model to a spatial local-to-unity model that exhibits weak mean reversion. We characterize the large sample behavior of regression inference with spatial I(1) variables, and establish that spurious regression is as much a problem with spatial I(1) data as it is with time series I(1) data. We develop asymptotically valid spatial unit root tests, stationarity tests, and inference methods for the local-to-unity parameter. Finally, we consider strategies for valid inference in regressions with persistent (I(1) or local-to-unity) spatial data, such as spatial analogues of first-differencing transformations.