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The paper deals with multivariate Gaussian random fields defined overgeneralized product spaces that involve the hypertorus. The assumption ofGaussianity implies the finite dimensional distributions to be completelyspecified by the covariance functions, being in this case matrix valuedmappings. We start by considering the spectral representations that in turnallow for a characterization of such covariance functions. We then provide somemethods for the construction of these matrix valued mappings. Finally, weconsider strategies to evade radial symmetry (called isotropy in spatialstatistics) and provide representation theorems for such a more general case.

Multivariate Gaussian Random Fields over Generalized Product Spaces involving the Hypertorus