This edition of Panoramas & Synthèses follows the session États de la Recherche : ‘Topics on compressible Navier-Stokes equations' that was organized from 21 to 25 May 2012 in the Laboratoire de Mathématiques - UMR5127 CNRS, 73376 Le Bourget du Lac, France. This national training session has been the opportunity to gather four internationally well known specialists (namely R. Danchin, B. Desjardins, A. Novotny, M. Perepetlisa) allowing to present the major actual mathematical developments related to the well-posedness character problem for the compressible Navier-Stokes equations to non-subject specialists. For the sake of unity, we have decided to collect in this special issue only the contributions dedicated to the non-degenerate viscosities case, hoping by this way to present a self-contained contribution on the subject : global weak-solutions à la Leray, intermediate solutions à la Hoff and strong solutions in critical spaces à la Fujita-Kato.