This work mainly concerns with the recently proposed Onsager-Burnett equations [Singh, Jadhav, and Agrawal, Phys. Rev. E 96, 013106 (2017)] for rarefied gas flows, and the progress achieved so far by solving these equations for some benchmark flow problems. Unlike the conventional Burnett equations, the OBurnett equations form a stable and thermodynamically-consistent set of higher- order continuum transport equations. Further, noticeable absence of higher-order derivatives in the OBurnett constitutive relations for stress tensor and heat flux vector renders the equations needing the same number of boundary conditions as that of the Navier-Stokes equations. These two important aspects: thermodynamic consistency and no need of additional boundary conditions, helps to set the OBurnett equations apart from the rest of the higher-order continuum theories. Available results of OBurnett equations for benchmark problems like force-driven plane Poiseuille flow and normal shocks are promising and helps to establish the validity of the equations. The recently proposed Grad's second problem and its solution within the Burnett hydrodynamics is also discussed at length and some important remarks are made in this context.