The following chapters present an overview of combustion and of CFD (Computational Fluid Dynamics) for combustion. The objective is not to repeat classical textbooks on these topics [379; 306; 288; 334; 340] but to focus on the place of instabilities in reacting flows and in CFD for reacting flows. These instabilities are found at many levels: Instabilities exist in individual flame fronts and lead to the formation of cells and of various unstable modes depending on molecular transport of chemical species and heat [379; 266]. Like any shear flow, reacting flows are submitted to hydrodynamic modes [273; 297] and to vortex formation. Acoustics play a major role in reacting flows: by coupling with heat release, they are the source of a major problem in many combustion devices: combustion instabilities [379; 340] which can induce high vibration levels and, in extreme cases, destroy combustion hardware in a few seconds. Instabilities are present in the physical problem studied but they are also present in the numerical methods used to simulate these mechanisms. Most high-fidelity numerical schemes required for Computational Fluid Dynamics exhibit low dissipation and therefore multiple non-physical instabilities (wiggles) arise which can require significant efforts to be kept under control [374; 362; 340]. Finally, CFD for reacting flows are performed today on massively parallel machines: these architectures coupled with centered schemes for turbulent flows lead to an additional type of instability linked to the growth of rounding errors and to a new type of instability where the solution depends on unexpected parameters such as the commutativity errors of addition, the initial condition or the number of processors.