The Auger Engineering Radio Array (AERA) is located at the Pierre Auger Observatory in Mendoza Province, Argentina. More than 150 autonomous antenna stations, spread over 17 km$^2$, are used to measure the radio emission from extensive air showers initiated by cosmic rays with energies above 0.1 EeV in the frequency range of 30 – 80 MHz. AERA is operated in coincidence with the other detectors of the Observatory, which comprise an array of water-Cherenkov detectors, fluorescence telescopes and buried scintillators that are used to detect muons. This gives a unique opportunity of cross-calibration between the detector types and complementary analyses of shower parameters. From the radio measurements we reconstruct cosmic-ray properties like energy, arrival direction and estimators of the mass composition, in particular the atmospheric depth $X_\mathrm{max}$ of maximum shower development. To determine $X_\mathrm{max}$, we follow a top-down approach with detailed simulations down to the individual particle level, also considering the refractive index. Two independent implementations are followed, both using an atmospheric model based on GDAS data. The results of the two analyses are consistent. We achieve a resolution of the radio detectors of ${\sim}39$ g/cm$^2$. Since radio measurements are solely sensitive to the electromagnetic part of the shower, we can measure the electron-muon ratio in combination with the muon detector of AMIGA as an additional mass estimator. Recently AERA demonstrated that for inclined showers the area of the radio-emission footprint extends to several square kilometers. This shows great potential for future large-scale radio arrays, since the large footprints allow for a wide spacing of the antennas.