The Navier–Stokes-Fourier (NSF) equations, the cornerstone of continuum fluid mechanics, have been remarkably successful in describing fluid flow at macroscopic scales. However, their accuracy deteriorates in regimes where molecular non-equilibrium effects become significant, characterized by a finite Knudsen number. This chapter provides a comprehensive review of the Chapman-Enskog method, a fundamental asymptotic technique in kinetic theory that bridges the microscopic Boltzmann equation and macroscopic continuum equations. We systematically derive the NSF equations as the first-order approximation of this expansion. The discussion then focuses on the second and third-order approximations, which yield the Burnett and super-Burnett equations, respectively. These higher-order equations introduce complex stress and heat flux terms dependent on higher-order gradients of flow variables, theoretically enabling the description of rarefaction effects such as velocity slip and temperature jump. We critically analyze the mathematical structure, physical significance and inherent challenges of these equations, including their notorious linear instability and the closure problem for boundary conditions.