The application of the Gaussian moment closure to micron-scale flows is considered. The mathematical formulation of the closure is reviewed as well as an extension to allow for diatomic gases and the treatment of solid-wall boundaries. The eigensystem of the set of coupled hyperbolic partial differential equations resulting from the moment closure is also reviewed. A parallel upwind finite-volume scheme with block-based adaptive mesh refinement (AMR) using Roe- and HLLE-type flux functions is developed for the numerical solution of the moment equations. Comparisons are made between numerical solutions of the Gaussian closure, other numerical solutions, and analytical solutions for several test problems, including Couette, boundary layer and cylinder flow, and under-expanded free jet flow, over a range of Knudsen numbers in order to demonstrate the applicability of the closure for a range of micron-scale flows. Agreement between previous results and the numerical solutions of the Gaussian closure for these problems are very encouraging.