Understanding the physics of supersonic combustion is the key to design a performing engine for scramjet-powered vehicles. Despite studies on supersonic combustion dating back to the 1950s, there are still numerous uncertainties and misunderstandings on this topic. The following questions need to be answered: How does compressibility affect mixing, flame anchoring, and combustion efficiency? How long must a combustor be to ensure complete mixing and combustion while avoiding prohibitive performance losses? How can reacting turbulent and compressible flows be modeled? Experimental results in the past have shown that supersonic combustion of hydrogen and air is feasible and takes place in a reasonable distance, which is a necessary requirement in actual hypersonic vehicles powered by supersonic combustion ramjets. These results are explained based on a theoretical analysis of the physical mechanisms driving mixing and combustion in supersonic airstreams, where they are found to be different from those in the incompressible regime. In particular, the classic Kolmogorov scaling is shown to be no longer strictly valid, and the flame regime is predicted to be significantly affected by compressibility and different from that of subsonic flames. This analysis is also supported by the results of the numerical simulations presented, showing that by generating sufficiently intense turbulence, a supersonic combustion flame is short and can indeed anchor within a small distance from fuel injectors, with the flame typically burning in the so-called flamelets-in-eddies regime.