The flow in an aero engine combustor is characterized by high levels of turbulence in a high-pressure two-phase air-kerosene reacting flow, with intensive heat transfer. Moreover large- scale structures induced by the swirl stabilizing processes generate strong unsteadiness for mixing and combustion that could couple with acoustics wave propagation and lead to combustion instabilities especially for lean combustors. This Lecture Series will review the aero-engine combustor design requirements, describe the understanding and the tools available for the analysis of fuel injection and spray evaporation, chemistry and physics of combustion, aerodynamics, mixing and unsteadiness, and heat transfer and cooling. The use of Large Eddy Simulation will be the subject of specific lectures. Operability will be addressed in particular with regards to operation under unsteady conditions, ignition and altitude relight. Emphasis will be put on lean combustion that has a low NOx emission potential, and the experimental demonstration of low NOx technologies in a representative environment will be presented. As a conclusion, perspectives opened by the use of alternative fuels will be discussed.