Minimal surfaces are critical points of the area functional, and have been of great interest to differential geometers since the time of Euler and Bernoulli. Their study usually involves situations where we fix some target manifold and ask whether it contains minimal surfaces of certain kinds, and if so how many, and so on. In this talk, we will approach things from the opposite direction: given some surface, can it be realised as a minimal surface in some ambient space? When we insist that the ambient space is hyperbolic, it's possible to obtain an elegant answer to this question via the concept of a formal minimal surface. After introducing these, we will attempt to say something about global aspects of their geometry, some recent results and open problems.