Minkowski space of dimension 2+1 is the Lorentzian analogue of Euclidean 3-space. It is well-known that there exists an isometric embedding of the hyperbolic plane in Minkowski space, which is the analogue of the embedding of the round sphere in Euclidean space. However, differently from the Euclidean case, the embedding of the hyperbolic plane is not unique up to global isometries. In this talk I will discuss several results on the classification of these embeddings, and explain how this problem is related to Monge-Ampère equations, harmonic maps, and Teichmüller theory. This is joint work with Francesco Bonsante and Peter Smillie.