This paper presents recent progress in the field of thermoacoustic combustion instabilities in propulsion engines such as rockets or gas turbines. Combustion instabilities have been studied for more than a century in simple laminar configurations as well as in laboratory-scale turbulent flames. These instabilities are also encountered in real engines but new mechanisms appear in these systems because of obvious differences with academic burners: larger Reynolds numbers, higher pressures and power densities, multiple inlet systems, complex fuels. Other differences are more subtle: real engines often feature specific unstable modes such as azimuthal instabilities in gas turbines or transverse modes in rocket chambers. Hydrodynamic instability modes can also differ as well as the combustion regimes, which can require very different simulation models. The integration of chambers in real engines implies that compressor and turbine impedances control instabilities directly so that the determination of the impedances of turbomachinery elements becomes a key issue. Gathering experimental data on combustion instabilities is difficult in real engines and Large Eddy Simulation (LES) has become a major tool in this field. Recent examples, however, show that LES is not sufficient and that theory, even in these complex systems, plays a major role to understand both experimental and LES results and to identify mitigation techniques.