Turbulence influences the properties of astrophysical fluids on a wide range of scales, from protoplanetary disks to the outskirts of galaxy clusters, and governs the critical process of star formation in galaxies like the Milky Way. While the theory of incompressible, subsonic turbulence benefits from a rich history spanning more than half a decade, the theory of supersonic turbulence in astrophysical contexts remains work in progress. Aided by large-scale numerical simulations, much of the theoretical success in modeling supersonic isothermal turbulence involves a coarse description of its statistical properties. After reviewing the theory of turbulence and its relevance for astronomical observations, I will present a fundamentally new framework for thinking about supersonic turbulence that focuses on describing the properties of shocked, dense regions. This new approach benefits from a direct connection with the astrophysics of star formation, and makes verifiable predictions for the structure of star forming clouds. I will conclude by discussing new computational methods engineered to tackle problems like supersonic turbulence in astrophysical contexts.