Minkowski space of dimension 2+1 is the Lorentzian analogue of Euclidean 3-space. It is well-known that the Gauss map of a Riemannian surface of constant mean curvature (CMC) in Minkowski space is harmonic, while the Gauss map of a surface of constant Gaussian curvature (CGC) is minimal Lagrangian. In this talk I will present a classification result for properly embedded CMC and CGC surfaces in Minkowski space, and show how harmonic maps from the complex plane to the hyperbolic plane play an essential role in the proof.