QCD at finite temperature and chemical potential from Dyson-Schwinger equations

We review results for the phase diagram of QCD, the properties of quarks and gluons and the resulting properties of strongly interacting matter at finite temperature and chemical potential. The interplay of two different but related transitions in QCD, chiral symmetry restoration and deconfinement, leads to a rich phenomenology when external parameters such as quark masses, volume, temperature and chemical potential are varied. We discuss the progress in this field from a theoretical perspective, focusing on non-perturbative QCD as encoded in the functional approach via Dyson-Schwinger and Bethe-Salpeter equations. We aim at a pedagogical overview on the physics associated with the structure of this framework and explain connections to other approaches, in particular with the functional renormalization group and lattice QCD. We discuss various aspects associated with the variation of the quark masses, assess recent results for the QCD phase diagram including the location of a putative critical end-point for $N_f=2+1$ and $N_f=2+1+1$, discuss results for quark spectral functions and summarise aspects of QCD thermodynamics and fluctuations.