Double categories are a flexible 2-dimensional setting that allows us to encode two types of morphisms between objects, as well as a notion of higher cells. Surprisingly, unlike most categorical structures, there is no canonical notion of "equivalence of double categories", as every possible definition requires us to make a choice. In this talk we will introduce double categories and illustrate the issue that arises when defining equivalences. Then, we will show how we can use homotopy theory to give a decisive answer as to who the "canonical double categorical equivalences" should be. In the process, we will show how to construct a plethora of model structures on double categories whose homotopy theories encode different 2-dimensional structures. Based on joint work with Lyne Moser and Paula Verdugo.