The global existence issue for the isentropic compressible Navier-Stokes equations in the critical regularity framework has been addressed by R. Danchin more than fifteen years ago. However, whether (optimal) time-decay rates could be shown in general critical spaces and any high dimensions has remained an open question. In this talk, we survey some recent results on the optimal time-decay estimates not only in the L^2 critical framework but also in the more general L^p critical framework.
