This thesis is devoted to the numerical investigation of mixing and non- premixed combustion of cryogenic propellants at supercritical pressures. These severe conditions are commonly encountered in high pressure combustion chambers, such as those of liquid-fueled rocket engines (LRE), and lead to significant deviations from the ideal gas thermodynamic behavior of the reacting mixtures. The non-premixed laminar flame structure of liquid oxygen (LOx) and methane or liquid natural gas (LNG) mixtures, a recently proposed LRE propellants com- bination, is investigated by means of a general fluid unsteady flamelet solver. Real gas effects are analyzed on prototypical unsteady flame phenomena such as autoignition and re-ignition/quenching caused by strain perturbations. Such effects influence different flame regions depending on pressure, as well as the critical strain values that a laminar flame can sustain before quenching occurs. Moreover the flame structure is also influenced by the composition of the LNG, in particular the early stage soot precursors production and oxidation. In order to shed light on real gas mixing, a low-Mach approximation for real gas reacting mixtures is presented. A single species non-reacting real gas model is implemented in a highly scalable spectral element computational fluid dynamic (CFD) code with state of the art thermodynamic and transport properties. Transcritical and supercritical planar temporal jets, are chosen as representative test cases for investigating high-pressure mixing by means of direct numerical simulations. The pseudo-boiling phenomenon, occurring in transcritical flows, significantly influences the jet development, mitigating the development of shear layer instabilities and leading to a liquid-like jet break-up. Moreover pseudo-boiling is confined in a narrow spatial region suggesting particular care in the turbulent combustion modeling of non-premixed flames when transcritical thermodynamic conditions are encountered. The results of the present thesis, its physical insights as well as the modeling considerations involved, can be of support in the development of future CFD tools capable of simulating real engine operative conditions and configurations.