We set up a general framework for modeling non-Gaussian multivariate stochastic processes by transforming underlying multivariate Gaussian processes. This general framework includes multivariate spatial random felds, multivariate time series, and multivariate spatio-temporal processes, whereas the respective univariate processes can also be seen as special cases. We advocate joint modeling of the transformation and the cross-/auto-correlation structure of the latent multivariate Gaussian process, for better estimation and prediction performance. We provide two useful models, the Tukey g-and-h transformed vector autoregressive model and the sinh-arcsinhtransformed multivariate Matérn random feld. We evaluate them with a simulation study. Finally, we apply the two models to a wind data set for modeling the two perpendicular components of wind speed vectors. Both the simulation study and data analysis show the advantages of the joint modeling approach.