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We propose a new model for regression and dependence analysis when addressing spatial data with possibly heavy tails and an asymmetric marginal distribution. We first propose a stationary process with t marginals obtained through scale mixing of a Gaussian process with an inverse square root process with Gamma marginals. We then generalize this construction by considering a skew‐Gaussian process, thus obtaining a process with skew‐t marginal distributions. For the proposed (skew) t process, we study the second‐order and geometrical properties and in the t case, we provide analytic expressions for the bivariate distribution. In an extensive simulation study, we investigate the use of the weighted pairwise likelihood as a method of estimation for the t process. Moreover we compare the performance of the optimal linear predictor of the t process versus the optimal Gaussian predictor. Finally, the effectiveness of our methodology is illustrated by analyzing a georeferenced dataset on maximum temperatures in Australia. [ABSTRACT FROM AUTHOR]