The QCD phase diagram at finite densities

I discuss recent progress towards calculating the QCD phase diagram at finite density using the functional Renormalisation Group (fRG). After introducing the fRG as applied to QCD, I explain some of the challenges encountered in functional approaches to the QCD phase diagram.
Many of these can be resolved by recent developments of new numerical methods. In particular, the application of numerical hydrodynamics to RG flows and resolution of momentum dependences allow us to make progress towards quantitative access to the region of the conjectured critical end-point (CEP) of the QCD phase diagram.
An interesting result is the appearance of new phases characterised by spatial modulations (the moat regime) and inhomogeneous condensates at high densities from a self-consistent first-principles calculation. For the near future, a clear program emerges to further pinpoint the CEP and its possibly modified nature using the fRG.