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New, high-fidelity numerical methods are required in order to help design the next generation of high-performance propulsion systems and aircraft. In particular, these realworld applications often involve highly-unsteady, turbulent, compressible flows which are difficult to simulate using traditional methods. Space-time methods offer a promising approach for accurately and efficiently simulating these flows. Unfortunately, space-time methods require four-dimensional (4D) meshes. Thus far, limited efforts have been devoted towards generating these meshes. The purpose of this dissertation is to begin addressing this issue. Towards this end, a novel method is introduced for generating constrained hypervolume boundary meshes in 4D space time. The method begins by dividing the space-time domain into time slabs, where hypersurface meshes are generated separately for each slab. For a particular slab, a hypervolume mesh is generated from the points associated with the hypersurface mesh using a Delaunay-based point insertion algorithm. Next, the hypersurface boundary facets are recovered using a unique set of pentatope decompositions designed for segment, face, and facet recovery. Lastly, a unique set of 4D bistellar flips and 4D quality heuristics designed for mesh quality improvement are discussed. Numerical results are presented for each aspect of this method to demonstrate its validity