The Cauchy problem for wave maps on hyperbolic space exhibits several features of a different nature than the corresponding problem on flat space. In this talk we'll focus on the question of gauge choice and we'll sketch the proof of small data global well-posedness and scattering in high dimensions using the moving frame approach introduced by Shatah and Struwe. In this setting of a curved domain, the argument will rely crucially on the fact that the main dynamic equations in Tao's caloric gauge are scalar, rather than the tensorial equations that arise in say, the Coulomb gauge. This talk is based on joint work with Sung-Jin Oh and Sohrab Shahshahani.