The Burgers Equation: A Century of Nonlinearity, from Turbulence to Machine Learning -- A Comprehensive Review

The Burgers equation is the simplest model that captures the interplay between nonlinear advection and dissipation in a continuum. Since its introduction in the 1940s as a toy model of turbulence, it has become a cornerstone of applied mathematics, statistical physics, and computational science. Its exact solvability through the Hopf-Cole transformation, its role as the canonical prototype of shock formation and entropy solutions in conservation laws, and its connection to the Kardar-Parisi-Zhang universality class make it an ideal laboratory for both theory and numerics. This review provides a comprehensive and critical assessment of nearly a century of research on the Burgers equation. We trace its historical origins, survey the analytical theory of the viscous and inviscid equations, and examine the stochastic Burgers equation and its deep ties to surface growth and interacting particle systems. We then analyze the rich phenomenology of Burgers turbulence, including shock statistics, intermittency, and anomalous scaling. Numerical methods, from classical finite-difference schemes to spectral, finite-element, and structure-preserving approaches, are evaluated in the context of the equation's characteristic shock structures. We also discuss fractional, generalized, and multi-dimensional extensions, and the broad spectrum of applications in traffic flow, nonlinear acoustics, cosmology, and granular media. Particular attention is given to the recent surge of machine-learning approaches, including physics-informed neural networks and neural operators, and to the newest results published in 2025 and 2026. Finally, we identify persistent open problems and chart future research directions at the interface of analysis, computation, and data-driven modeling.