There is a great demand for analyzing multivariate measurements observed across space and over time, due to an increasing wealth of multivariate spatial, temporal, or spatio-temporal data, which may be treated as the realizations of vector (multivariate) random fields. As one of the most important random fields in theory and application, Gaussian random field has been extensively investigated in the literature. Non-Gaussian models and random fields are often encountered in many natural and applied science areas, with specific reasons for assuming particular non-Gaussian finite-dimensional distributions in practice. One of the objectives of this dissertation is to introduce a new non-Gaussian vector random field, which belongs to the family of elliptically contoured vector random fields. This new field is named the K-differenced vector random fields, because its finite-dimensional densities are the difference of two Bessel K functions. A K-differenced vector random field is of second-order and allows for any possible correlation structure, just as a Gaussian one does. It includes a Laplace vector random field as a limiting case. This dissertation studies the properties of the K-differenced vector random field and proposes some covariance matrix structures for not only a K-differenced vector random field but also a second-order elliptically contoured one. Other objectives of this dissertation are to construct the K-differenced random variable or random vector as the scale mixture of normal random variables or vectors and to derive its density and characteristic functions. Simulations of the K-differenced distribution have been made through Monte Carlo procedures. Maximum likelihood estimators of the parameters for the simulations are found numerically via MatLab.