The Auger Engineering Radio Array (AERA), as part of the Pierre Auger Observatory, is an array of $153$ radio antennas spanning an area of $17$ km$^2$, currently the largest of its kind, that probes the nature of ultra-high-energy cosmic rays at energies around the transition from Galactic to extra-galactic origin. It measures the MHz radio emission of extensive air showers produced by cosmic rays hitting our atmosphere. We show the recent work by AERA, such as the measurement of the muon content of inclined air showers and the stability of the measured radio signal over almost a decade, as measured with the Galactic radio background. In particular, we highlight the measurements of the depths of the shower maxima $X_\mathrm{max}$, which we use to make inferences about the mass composition of cosmic rays. We reconstruct $X_\mathrm{max}$ by comparing the measured radio footprint on the ground to an ensemble of footprints from Monte-Carlo CORSIKA/CoREAS air shower simulations. We compare our $X_\mathrm{max}$ reconstruction with fluorescence $X_\mathrm{max}$ measurements on a per-event basis, a setup unique to the Pierre Auger Observatory, and show the methods to be fully compatible. We determine the resolution of our method as a function of energy and reach a precision better than $15$ g cm$^{-2}$ at the highest energies. With a bias-free set of around $600$ showers, we find agreement with the Auger fluorescence measurements at energies between $10^{17.5}$ to $10^{18.8}$ eV.