Combustion in the supersonic regime presents several challenges when compared to its low-speed counterpart. Here we review some of these challenges, and we describe some of the key features of one of the canonical flow fields in supersonic combustion: the reacting transverse jet in a supersonic crossflow (JISCF). From a practical standpoint, the key challenges that limit the control of this combustion regime are fast mixing, robust flame holding and stability. In turn, these aspects are controlled by the complex effects introduced by chemistry, compressibility, shocks and shock/flow interactions, turbulence and the underlying coupling among them. Some of their properties are discussed here. In particular, for a JISCF in a Mach 2.4 high enthalpy crossflow, the reaction zone structure, its dependence on near-wall events, boundary layer, and shock/boundary layer interaction are described. We demonstrate the paramount importance of the coupling between boundary layers and compressibility to provide mechanisms for flame stabilization at the wall. Mixing characteristics, overall structure, and the link to global parameters (momentum flux, velocity and density ratios) that characterize the JISCF, and possibly free shear supersonic flows in general, is also highlighted from non-reacting experiments. Nomenclature d = Jet exit diameter J = Jet-to-crossflow momentum flux ratio, ρ jet U 2 jet ρ a U 2 a M = Mach number p = Pressure p j,o = Jet stagnation pressure Re d = Injector jet exit Reynolds number, ρ jet U jet d µ jet r = Velocity ratio, U jet /U a s = Density ratio, ρ jet /ρ a T = Temperature U = Speed x = Streamwise direction y = Wall-normal direction z = Spanwise direction Greek letters: µ = Dynamic viscosity ρ = Density Subscript: a = Freestream conditions