<p>Physics-based models play a central role in the quantitative understanding and simulation of lithium–ion batteries. Among them, the Doyle–Fuller–Newman (which we will refer to as DFN) model, also known as the pseudo-two-dimensional (P2D) model, has become the reference continuum framework for describing the coupled transport, electrochemical, and kinetic phenomena governing cell operation. While widely used in both academia and industry, the mathematical structure of the DFN model, its underlying assumptions, and the numerical strategies employed for its solution are often presented in a fragmented or application-driven manner. This review provides a coherent and mathematically oriented overview of the DFN framework, with particular emphasis on its multiscale formulation, governing equations, and numerical implementation. We first summarise the physical and electrochemical principles underlying lithium–ion cell operation and show how these mechanisms are translated into the DFN model through porous-electrode theory. We then present the resulting coupled system of partial differential equations, highlighting its multiscale structure and nonlinear couplings. A dedicated part of the review is devoted to numerical solvers and implementations for DFN-type models. We discuss representative finite-element and finite-volume approaches, including COMSOL Multiphysics, DandeLiion, and PyBaMM, and analyse how solver design choices influence robustness, accuracy, and behaviour near limiting regimes. A comparative numerical study illustrates how different treatments of stiffness and constraints can lead to solver-dependent predictions close to end-of-discharge. By emphasising the interplay between physical modelling assumptions and their mathematical and numerical consequences, this work provides applied mathematicians and numerical analysts with a unified perspective on DFN-based battery modelling.</p>