This paper reviews the fundamentals of conditional moment closure (CMC) methods for the prediction of turbulent reacting flows, with particular emphasis on combustion. It also surveys several of the applications that have been made. CMC methods predict the conditional averages and higher moments of quantities such as species mass fractions and enthalpy, conditional on the mixture fraction or reaction progress variable having a particular value. A brief introduction is given to generalized functions and probability density function (pdf) methods. This is followed by an exposition on the various methods of derivation for the CMC equation and the general characteristics of this equation and its boundary conditions. Simplifications that can be made in slender layer flows such as jets and plumes are outlined and examples of application of the technique to such flows are given. The method allows the definition of a new class of simplified reactors related to the well known perfectly stirred reactor and plug flow reactor: these are outlined. CMC predictions are compared to experiment and direct numerical simulations for flows with homogeneous turbulence. Derivation and modeling of the equations for conditional variances and covariances are outlined and their use in second-order CMC illustrated. Brief review is made of progress on application of the method to problems involving differential diffusion, multiple conditioning, sprays and premixed combustion.