Transcritical and supercritical flows are encountered in cryogenic rocket engines, where reactants are stored and injected at high pressure and low temperature. This implies specific thermodynamic and diffusive behaviors, which have direct impacts on the combustion and flow dynamics. To predict such systems, the Large Eddy Simulation approach is today commonly used as it allows a good description of the turbulence and other transient phenomena. The formulation involves the fully compressible flow equations, associated to a cubic equation of state and modified transport properties to properly describe the dense fluid. In the paper, the complete formulation is recalled and specific numerical treatments implemented in the code AVBP to ensure accuracy and robustness are described. Validation test cases as well as rocket engine applications are then presented to demonstrate the capacities of the numerical approach and the reliability of the models for such complex systems. Ongoing developments and future works are finally given.