The problem of finding a surface in 3-dimensional Euclidean space with a given first fundamental form was classically known as Minding’s problem, and is closely related to the problem of determining what surfaces in space are `applicable’ (i.e., isometric without being congruent) to a given surface. We review classical approaches to these problems, including coordinate-based approaches exposed by Darboux and Eisenhart, as well as the moving-frames based approaches used by Cartan. In particular, the first approach involves reducing Minding’s problem to solving a single PDE; settling the question of when this PDE is integrable by the method of Darboux is surprisingly easy, once we answer the analogous question for the isometric embedding system arising from the moving frames approach. The latter question is settled in recent joint work Jeanne Clelland and Peter Vassiliou.