In this talk I will review a number of recent developments in perturbative heterotic string compactifications and two-dimensional (0,2) theories in particular. I'll begin with a general overview of chiral ring structures in two dimensions, including quantum sheaf cohomology, the heterotic analogue of quantum cohomology. I'll then turn to dualities in 2d (0,2) theories, and examine the recent Gadde-Gukov-Putrov triality as a particularly interesting case that, unlike most others, cannot be simply understood solely in geometry. In the the last part of the talk I will discuss recent checks of triality via chiral ring computations, joint with Bei Jia and Jirui Guo.