What are modern approaches to describe and/or quantify aromaticity?
I'm more familiar with the Electron Localization Function approach to aromaticity commented by Quantum AMERICCINO, it's been proposed that the average value of the $\text{ELF}_{\sigma}$ and $\text{ELF}_{\pi}$ bifurcations would be used for a general scale of aromaticity of a molecule or molecular cluster. The points of bifurcation can be interpreted as a measure of the interaction among the different basins or, simply, electron delocalization. Although the ELF depends on the density (i.e. no separate information of $\pi$ or $\sigma$ bond), a topological analysis on a separated ELF formed only by the $\pi$ orbitals and other formed by the $\sigma$ orbitals can be done (also you can do a separation for the spin $\alpha$ and $\beta$ contributions to the ELF). [J. Chem. Theory Comput. 2005, 1 (1), 83–86.]

There is another approach based on how a bond responds when its electron density is perturbed by a finite amount. Specifically, they used the interaction coordinate (IC) to see this. Quoting


  If the delocalization in the aromatic system is more extensive, for constraining a bond by unit displacement from the equilibrium geometry, it requires more energy. So the response of the other internal coordinates will be greater for constrained optimization and is proportional to the effective delocalization and hence related to the aromatic stabilization. The IC is a measure of the response of a bond (its electron density) for constrained optimization when another bond or angle is stretched by one unit (its electron density is perturbed by a measured amount).


Instead of taking the full molecule, they take only the aromatic skeleton to account the electron density perturbation on the system. Also, it says that this is very general yet, so the IC approach can take new directions. 
[J. Phys. Chem. A 2016, 120 (18), 2894-2901.]

Another approach is making a relationship between the amount of the electron density (due to ring size, number of $\pi$ electrons, etc) and the non-nuclear magnetic shielding (cf. NICS). 
[Phys. Chem. Chem. Phys. 2010, 12 (39), 12630–12637.]