Probability density function (PDF) methods offer compelling advantages for modeling chemically reacting turbulent flows. In particular, they provide an elegant and effective resolution to the closure problems that arise from averaging or filtering the highly nonlinear chemical source terms, and terms that correspond to other one-point physical processes (e.g., radiative emission) in the instantaneous governing equations. This review is limited to transported PDF methods, where one models and solves an equation that governs the evolution of the one-point, one-time PDF for a set of variables that determines the local thermochemical and/or hydrodynamic state of a reacting system. Progress over the previous 20-25 years (roughly since Pope's seminal paper [24[) is covered, with emphasis on developments over the past decade. For clarity and concreteness, two current mainstream approaches are adopted as baselines: composition PDF and velocity-composition PDF methods for low-Mach-number reacting ideal-gas mixtures, with standard closure models for key physical processes (e.g., mixing models), and consistent hybrid Lagrangian particle/Eulerian mesh numerical solution algorithms. Alternative formulations, other flow regimes, additional physics, advanced models, and alternative solution algorithms are introduced and discussed with respect to these baselines. Important developments that are discussed include velocity-composition-frequency PDF's, PDF-based methods as subfilter-scale models for large-eddy simulation (filtered density function methods), PDF-based modeling of thermal radiation heat transfer and turbulence-radiation interactions, PDF-based models for soot and liquid fuel sprays, and Eulerian field methods for solving modeled PDF transport equations. Examples of applications to canonical systems, laboratory-scale flames, and practical combustion devices are provided to emphasize key points. An attempt has been made throughout to strike a balance between rigor and accessibility, between breadth and depth of coverage, and between fundamental physics and practical relevance. It is hoped that this review will contribute to broadening the accessibility of PDF methods and to dispelling misconceptions about ME methods. Although PDF methods have been applied primarily to reacting ideal-gas mixtures using Single-turbulence-scale models, multiple-physics, multiple-scale information is readily incorporated. And while most applications to date have been to laboratory-scale nonpremixed flames, PDF methods can be, and have been, applied to high-Damkohler-number systems as well as to low-to-moderate-Damkohler-number systems, to premixed systems as well as to nonpremixed and partially premixed systems, and to practical combustion devices as well as to laboratory-scale flames. It is anticipated that PDF-based methods will be adopted even more broadly through the 21st century to address important combustion-related energy and environmental issues. (C) 2009 Elsevier Ltd. All rights reserved.