When Yuri Gagarin was launched into orbit in 1961, the probability of a rocket blowing up on take-off was around 50%. In those days, one of the most persistent causes of failure was a violent oscillation caused by the coupling between acoustics and heat release in the combustion chamber. If more heat release than average occurs during moments of high pressure and less heat release than average occurs during moments of low pressure then, over a cycle, more work is done during the expansion phase than is absorbed during the compression phase, causing oscillations to grow. These thermoacoustic oscillations have caused countless rocket engine and gas turbine failures since the 1930s and have been studied extensively. Nevertheless, they are still one of the major problems facing rocket and gas turbinemanufacturers today. The ultimate goal of rocket and gas turbine manufacturers is to eliminate or control thermoacoustic oscillations, either through feedback control or passive control. Feedback control works well in simple thermoacoustic systems but is challenging in industrial systems because the sensors and actuators have to withstand very harsh environments. Furthermore, feedback control is unacceptably risky insome applications, such as aircraft. For these reasons, passive control is preferable, either by good initial design, or by adding a passive device to an existing system. In order to control a thermoacoustic system passively, it is necessary to understand why the system oscillates. It is well known that acoustic perturbations to the velocity or pressure cause heat release perturbations some time later, and that these lead to the feedback loop described above. Other mechanisms, such as the reflection of entropy waves at a sonic throat, are also known. However, experiments show that even small changes to a system can significantly alter its stability, showing that the details of these processes are very influential. The aims of this course are: to describe how thermoacoustic oscillations arise, to show the flame dynamics that cause fluctuating heat release, to show how these details are uncovered through experimental measurements, to introduce linear and nonlinear methods of analysis, to introduce methods that can reveal which details of a thermoacoustic system are most influential, and to give examples of these processes in industrial thermoacoustic systems. The course is aimed at doctoral students in an early stage of a PhD in Thermoacoustics; researchers with a background in flow stability who are interested in a new area; and practicing engineers in a closely-related area such as gas turbine or rocket engine research.