In the past decade the functional approach to 1st principles QCD has made rapid advances. By now the results for the chiral crossover temperature and observables such as quark condensates agree quantitatively with the respective lattice results for vanishing and low baryon chemical potential mu_B. The large chemical potential regime interesting for BES is notoriously difficult to resolve within QCD. While lattice simulations suffer from the sign problem for chemical potentials mu_B/T> 1-3, functional approaches are still qualitatively reliable in this regime, but face a rapidly increasing systematic error within the current approximations.
In the present talk I give an overview of the current state of the art of 1st principles functional approaches (functional renormalisation group, Dyson-Schwinger equations) to the phase structure of QCD at finite densities. The location of the chiral phase boundary at larger chemical potential and that of the potential critical end point, potential signatures in terms of fluctuations observables as well as transport properties are discussed. It is argued that the systematic improvements in functional approaches may allow for quantitative predictions at large densities (mu_B/T > 4) within the next few years.