  The classical Mat\'ern model has been a staple in spatial statistics. Novel
data-rich applications in environmental and physical sciences, however, call
for new, flexible vector-valued spatial and space-time models. Therefore, the
extension of the classical Mat\'ern model has been a problem of active
theoretical and methodological interest. In this paper, we offer a new
perspective to extending the Mat\'ern covariance model to the vector-valued
setting. We adopt a spectral, stochastic integral approach, which allows us to
address challenging issues on the validity of the covariance structure and at
the same time to obtain new, flexible, and interpretable models. In particular,
our multivariate extensions of the Mat\'ern model allow for asymmetric
covariance structures. Moreover, the spectral approach provides an essentially
complete flexibility in modeling the local structure of the process. We
establish closed-form representations of the cross-covariances when available,
compare them with existing models, simulate Gaussian instances of these new
processes, and demonstrate estimation of the model's parameters through maximum
likelihood. An application of the new class of multivariate Mat\'ern models to
environmental data indicate their success in capturing inherent
covariance-asymmetry phenomena.
