Spatio-temporal processes involving more than one variable emerge in various fields. Any serious attempt of statistical inference and prediction for multivariate data require knowledge about the dependency structures within and across variables. In this work, we provide general conditions leading to positive semi-definiteness of the overall matrix-valued covariance functions. Both the marginal and cross-covariance functions belong to a generally non-separable Matérn class spatio-temporal covariance functions, but with possibly different scale, smoothness and space–time separability parameters. The main focus of this work is on bivariate spatio-temporal random fields. As an illustration, the model is fitted on a set of bivariate air pollution data.