Agradecimentos: The authors express their sincere thanks to G. Denicol, J. Noronha, J. Noronha-Hostler, E. Fraga, G. Torrieri, T. Csörgo, Y. Sinyukov, H. Petersen, T. Osada, for invaluable suggestions and in particular to G. Denicol, J. Noronha, J. Noronha-Hostler, G. Torrieri and T. Csörgo for critical readings of the manuscript. We also acknowledge E. Bratkovskaya and W. Cassing for providing the PHSD code and as well as the support concerning the physics of the model. This work has been supported by CNPq, FAPERJ, CAPES, PRONEX and grant 2014/09167-8, São Paulo Research Foundation (FAPESP) Ver menos

Abstract: We review several facets of the hydrodynamic description of the relativistic heavy ion collisions, starting from the historical motivation to the present understandings of the observed collective aspects of experimental data, especially those of the most recent RHIC and LHC results. In this report, we particularly focus on the conceptual questions and the physical foundations of the validity of the hydrodynamic approach itself. We also discuss recent efforts to clarify some of the points in this direction, such as the various forms of derivations of relativistic hydrodynamics together with the limitations intrinsic to the traditional approaches, variational approaches, known analytic solutions for special cases, and several new theoretical developments. Throughout this review, we stress the role of course-graining procedure in the hydrodynamic description and discuss its relation to the physical observables through the analysis of a hydrodynamic mapping of a microscopic transport model. Several questions to be answered to clarify the physics of collective phenomena in the relativistic heavy ion collisions are pointed out Ver menos