The basic equations governing a turbulent reacting system are presented, starting with the unaveraged/unfiltered partial differential equations (PDEs) and ancillary equations for a gas-phase multicomponent reacting system. These equations describe both laminar flames and turbulent flames where all continuum spatial and temporal scales are fully resolved (direct numerical simulation—DNS). InÂ simulations of practical turbulent combustion systems, it is not feasible to resolve all relevant scales explicitly, and one of two approaches is usually adopted to reduce the dynamic range of scales and to account for influences of turbulent fluctuations at unresolved scales on the resolved scales: Reynolds averaging (Reynolds-averaged Navier–Stokes—RANS), in which the influences of all turbulent fluctuations with respect to an appropriately defined local mean are modeled, or spatial filtering (large-eddy simulation—LES) in which only the influences of turbulent fluctuations at scales smaller than a prescribed lower limit (the filter scale) are modeled. Probability density functions (PDFs) are a particularly effective approach for modeling chemically reacting turbulent flows with radiative heat transfer; PDF methods are introduced here, and are subsequently used in many of the examples that are discussed in later chapters. The general notion of turbulence–chemistry interactions (TCI) in RANS and in LES is discussed, using the PDF framework. Finally, extensions to accommodate multiphase systems (soot, liquid fuel sprays, and coal) are discussed. Together with the material on radiation and turbulence–radiation interactions (TRI) in the next chapter, this provides the foundation that is needed for the examples that are presented and discussed in subsequent chapters.