This course is intended to provide the background and technical foundation for understanding combustion instabilities in all types of combustion systems. Examples and case histories of problems in solid rockets and liquid-fueled systems provide practical motivations for the basic material covered. Much of the course is concerned with the interpretation of physical behavior using approximate methods of analysis for linear and nonlinear phenomena. The basic equations of motion are carefully derived in forms specially suited to treat combustion instabilities. Fundamentals of acoustics and adaptations to combustion systems are explained. Formulation and results of numerical simulations are discussed for solid rockets and liquid-fueled systems. Potential applications and experience with both passive and active control are thoroughly discussed and illustrated with examples.