Periodic acoustic motions in combustion chambers, a phenomenon commonly known as combustion instability, were discovered in air-breathing and rocket engines at about the same time in the late 1930s. Since then, unstable oscillations have occurred in most, if not practically all, new development programs. Indeed, because of the high density of energy release in volume having relatively low losses, conditions normally favor excitation and sustenance of oscillations in any combustion chamber intended for a propulsion system. This presentation will provide an overview of combustion instabilities in three types of propulsion systems (i.e., solid rocket motors, liquid rocket engines, and air-breathing engines), with emphasis focused on the state-of-the-art understanding and future research needs. Various research issues in acoustics, fluid mechanics, and chemistry related to oscillatory combustion in practical systems will be discussed. Both passive and active control techniques will be covered. In particular, the application of contemporary CFD schemes, approximate analytical methods, and experimental diagnostic tools to combustion instability studies will be addressed in detail.