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Multivariate Random Fields Most modern spatial datasets involve multiple variables that can exhibit complex cross-process dependencies. We review some classic approaches to building statistical models for multivariate spatial data, nearly all of which rely on specifying cross-covariance functions. We discuss an alternative viewpoint in the spectral domain that helps to illustrate fundamental limitations on existing models, and suggests insights into, and dangers of, popular constructions. The second half of the talk focuses on multivariate modeling for very large spatial datasets. We show that a multiresolution construction can approximate standard models, and retains computationally efficient algorithms for estimation and prediction.