In rarefied gas dynamics, classical continuum laws fail because collisions are too infrequent to maintain equilibrium, a situation reflected by large Knudsen numbers. Extended models based on kinetic‑theory moments introduce additional variables to capture non‑equilibrium effects, with the regularized 13‑moment system (R13) being a widely used example. Regularized moment equations now exist for various molecular models, mixtures, and polyatomic or granular gases, and higher‑order systems offer improved accuracy. Several numerical methods have been developed, including finite‑element schemes built on well‑posed variational formulations. The talk reviews recent advances in moment‑equation modeling and applications. Phase interfaces in rarefied or strongly non‑equilibrium regimes exhibit discontinuous jump conditions that couple interface motion and field variables, creating significant numerical challenges when combined with the R13 equations on evolving domains. Also, most existing R13 results remain confined to linearized settings, because fully nonlinear simulations are still difficult due to the limited robustness of standard entropy maximization. New nonlinear closures must be explored. Finally, adaptive modeling strategies—where higher‑order moment systems are activated only where needed—will be essential for efficient non-equilibrium simulations.