A comprehensive numerical and experimental study of normal shock waves in hypersonic axisymmetric jets of N$_2$ is presented. The numerical interpretation is based on the quasi-gasdynamic (QGD) approach, and its generalization (QGDR) for the breakdown of rotational–translational equilibrium. The experimental part, based on diagnostics by high-sensitivity Raman spectroscopy, provides absolute density and rotational temperatures along the expansion axis, including the wake beyond the shock. These quantities are used as a reference for the numerical work. The limits of applicability of the QGD approach in terms of the local Knudsen number, the influence of the computational grid on the numerical solution, the breakdown of rotation–translation equilibrium, and the possible formation of a recirculation vortex immediately downstream from the normal shock wave are the main topics considered.