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Using low-temperature cathodoluminescence spectroscopy, we study the properties of N- and Al-polar AlN layers grown by molecular beam epitaxy on bulk AlN{0001}. Compared to the bulk AlN substrate, layers of both polarities feature a suppression of deep-level luminescence, a total absence of the prevalent donor with an exciton binding energy of 28 meV, and a much increased intensity of the emission from free excitons. The dominant donor in these layers is characterized by an associated exciton binding energy of 13 meV. The observation of excited exciton states up to the exciton continuum allows us to directly extract the $\Gamma_{5}$ free exciton binding energy of 57 meV.