Book Description: Transported probability density function (PDF) methods have shown promise in modeling canonical flame configurations. More recently, they have been applied to simple engine configurations to demonstrate their feasibility in more practical applications, and to demonstrate the importance of accurate accounting for turbulence/chemistry interactions (TCI) in IC engines. The research conducted here advances the state-of-the-art in two areas: advanced physical models and numerical methods for multiphase chemically reacting turbulent flows; and advanced combustion systems for direct-injection diesel engines. The hypotheses that are tested in this thesis are that turbulent fluctuations significantly impact heat release and emissions in advanced diesel engines and that PDF methods capture TCI effects in real engines. Contributions to modeling and algorithms include: (1) liquid fuel spray/PDF coupling; (2) "real engine" applications of PDF particle tracking through complex meshes; and (3) modularization of coupling among detailed thermochemistry and PDF methods. Contributions to engine combustion have included: (1) multiple-cycle calculations; (2) modeled premixed/direct-injection splits; (3) insight into the roles of turbulence and TCI on autoignition and emissions; (4) detailed information from the PDF method including spatial non-homogeneity, fluctuation effects, NOx and soot prediction, and detailed speciation; and (5) quantitative comparisons between CFD and experiment for a real engine. This research shows that turbulence/chemistry interactions in real engine applications are important to understanding and modeling engine combustion. The work allows for better analysis and performance prediction of advanced engine designs and points to how the models may be improved. Already, the PDF model's application to the real engine case here has shown appreciably better robustness and quantitative accuracy in ignition and emissions predictions compared to a conventional finite-volume approach where TCI is not considered. Ignition timing, heat release, and emissions are captured favorably with the PDF method. Directions for future improvement and research are suggested to address the remaining modeling issues.