Hadronic matter at the edge: A survey of some theoretical approaches to the physics of the QCD phase diagram

In the past few years a wealth of high quality data has made possible to test current theoretical ideas about the properties of hadrons subject to extreme conditions of density and temperature. The relativistic heavy-ion program carried out at the CERN-SPS and under development at the BNL-RHIC and CERN-LHC has provided results that probe the evolution of collisions of hadronic matter at high energies from the initially large density to the late dilute stages. In addition, QCD on the lattice has produced results complementing these findings with first principles calculations for observables in a regime where perturbative techniques cannot describe the nature of strongly coupled systems. This work aims to review some recent developments that make use of field theoretical methods to describe the physics of hadrons at finite temperature and density. I concentrate on two of the main topics that have been explored in the last few years: (1) The search for the structure of the phase diagram and (2) analytical signals linked to the chiral symmetry restoration/deconfinement.