This paper defines double fibrations (fibrations of double categories) and describes their key examples and properties. In particular, it shows how double fibrations relate to existing fibrational notions such as monoidal fibrations and discrete double fibrations, proves a representation theorem for double fibrations, and shows how double fibrations are a type of internal fibration.

We give the definitions of model bicategory and q-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. We prove that the 2-dimensional localization of a model bicategory C at the weak equivalences can be computed as a bicategory Ho(C) whose 2-cells are classes of q-homotopies up to an equivalence relation.