A stabilized finite element framework for high-speed compressible flows is presented. The Streamline-Upwind/Petrov–Galerkin formulation augmented with discontinuity-capturing (DC) are the main constituents of the framework that enable accurate, efficient, and stable simulations in this flow regime. Full- and reduced-energy formulations are employed for this class of flow problems and their relative accuracy is assessed. In addition, a recently developed DC formulation is presented and is shown to be particularly well suited for hypersonic flows. Several verification and validation cases, ranging from 1D to 3D flows and supersonic to the hypersonic regimes, show the excellent performance of the proposed framework and set the stage for its deployment on more advanced applications.