The concept of birational divisor arises in different contexts; for example in work of V. V. Shokurov and, independently, P. Vojta. The aim of this talk is to survey related recent developments. Some emphasis will be placed on results that deal with Brauer groups of function fields of algebraic varieties (e.g., joint work with C. Ingalls). Time permitting, I hope also (to at least briefly) mention results which deal with complexity and distribution of rational points (e.g., extensions to the recent work of Ru and Vojta).