This study is concerned with the development of extended fluid-dynamic models for the prediction of micro-scale flows. When compared to classical fluid descriptions, such models must remain valid on scales where traditional techniques fail. Also, knowing that solution to these equations will be sought by numerical methods, the nature of the extended models must also be such that they are amenable to solution using computational techniques. Moment closures of kinetic theory offer the promise of satisfying both of these requirements. It is shown that the hyperbolic nature of moment equations imbue them with several numerical advantages including an extra order of spacial accuracy for a given reconstuction when compared to the Navier-Stokes equations and a reduced sensitivity to grid irregularities. In addition to this, the expanded set of parameters governed by the moment closures allow them to accurately model many strong non-equilibrium effects that are typical of micro-scale flows. Unfortunately, traditional moment models have suffered from various closure breakdowns, and robust models that offer a treatment for non-equilibrium viscous heat-conducting gas flows have been elusive. To address these issues, a regularized 10-moment closure is first proposed herein based on the maximum-entropy Gaussian moment closure. This mathematically wellbehaved model avoids closure breakdown through a strictly hyperbolic treatment for viscous effects and an elliptic formulation that accounts for non-equilibrium thermal diffusion. Moreover, steps toward the development of fully hyperbolic moment closures for the prediction of non-equilibrium viscous gas flow are made via two novel approaches. A thorough study of each of the proposed techniques is made through numerical solution of many classical flow problems.