Birational geometry plays an important role in string theory, principally because of how it controls moduli spaces. The ubiquity of toric varieties and their subvarieties in string compactifications thus motivates the study of birational geometry in the toric setting. In this talk, I will discuss some recent formal and computational advances in this direction, including how the Kahler moduli space of Kreuzer–Skarke Calabi–Yau threefolds extends beyond FRSTs and how the birational and enumerative geometry of toric subvariety Calabi–Yau threefolds can be extracted from simple toric computations. I will also discuss how birational data additionally facilitates the algorithmic computation of global sections of line bundles.