We overview recent progress in growth aspects of group III-nitride heterostructures for deep ultraviolet (DUV) light-emitting diodes (LEDs), with particular emphasis on the growth approaches for attaining high-quality AlN and high Al-molar fraction AlGaN. The discussion commences with the introduction of the current status of group III-nitride DUV LEDs and the remaining challenges. This segues into discussion of LED designs enabling high device performance followed by the review of advances in the methods for the growth of bulk single crystal AlN intended as a native substrate together with a discussion of its UV transparency. It should be stated, however, that due to the high-cost of bulk AlN substrates at the time of this writing, the growth of DUV LEDs on foreign substrates such as sapphire still dominates the field. On the deposition front, the heteroepitaxial growth approaches incorporate high-temperature metal organic chemical vapor deposition (MOCVD) and pulsed-flow growth, a variant of MOCVD, with the overarching goal of enhancing adatom surface mobility, and thus epitaxial lateral overgrowth which culminates in minimization the effect of lattice- and thermal-mismatches. This is followed by addressing the benefits of pseudomorphic growth of strained high Al-molar fraction AlGaN on AlN. Finally, methods utilized to enhance both p- and n-type conductivity of high Al-molar fraction AlGaN are reviewed.