In 1967, Gabriel and Zisman introduced the conditions needed on a class of arrows to admit a calculus of fractions. If one has a category C with a class W of weak equivalences that satisfy these conditions, the arrows in the homotopy category of fractions C[W-1] can be described as spans where the left leg is in W. This category is the universal category obtained from C by inverting the arrows in W.
In this talk I will discuss the bicategorical generalization of this: I will discuss the conditions needed for a bicalculus of fractions and give an explicit description of the 2-cells in C[W-1], the free bicategory obtained by turning the arrows in W into equivalences. This talk is based on my paper in Compositio Mathematica from 1996, but I will discuss some minor improvements on that paper as well as ways to generalize this further to monoidal bicategories of fractions, and eventually, tricategories of fractions.