Modeling and Numerical Simulation of Proton Exchange Membrane Fuel Cells: Concept, Methods, and Challenges provides a concise guide to the modelling of PEM fuel cells. The book offers detailed methodologies, codes, and algorithms on every aspect of PEM fuel cells, from cold start to degradation. The opening chapter of the book explains the development, basic principles, and components of PEM fuel cells, followed by the importance and objectives of PEM fuel cell modeling. The second chapter discusses the transport phenomena and mathematical formulation of macro-scale PEM fuel cell models, single cell and stack-level models, and model validation. The third chapter explains multi-phase transport modeling in PEM fuel cells, including different multiphase models including flow in gas flow channels, porous electrodes, and multi-phase model validation. It also addresses multiphase mixture formulation, finite-volume, direct numerical simulation, Lattice Boltzmann, and pore network models. The fourth chapter is dedicated to modeling the cold start process of PEM fuel cells, including the non-isothermal transient cold start model, reduced-dimensional transient model, and the impact of different parameters on the cold start performance. The fifth chapter covers the degradation and lifetime modeling of PEM fuel cells, including stress-induced degradation mechanisms, physics-based and data-driven modeling methods, and coupled performance-degradation models. Finally, the sixth chapter presents recent progress on multi-scale and multi-dimensional modeling of PEM fuel cells, including micro and nano-scale modeling and multi-scale coupled models. Modeling and Numerical Simulation of Proton Exchange Membrane Fuel Cells: Concept, Methods, and Challenges is a valuable reference for researchers, scientists, engineers, and professionals working on fuel cell technologies or interested in fuel cell modelling.