Abstract:We will survey recent developments relating to the Hull-Strominger system on non-Kahler Calabi-Yau threefolds. In particular, we will discuss the anomaly cancellation equation and its connection to Aeppli cohomology and D-bar operators. The Aeppli class approach will be compared to the Bott-Chern balanced class approach. Along the way, we will also consider aspects of the moduli problem for the system.