Negative curvature of Riemannian manifolds, originally a differentiable theory, has been discretized in several phases. The first phase might have been Dehn's algorithm for the word problem in a surface group; I am guessing that at the time this might have seemed more an "application" of hyperbolic geometry than a discretization of it. But then comes the next big phase, the development of small cancellation theory, in which Dehn's algorithm (and related tools) were applied to many abstractly defined groups. The culminating phase was the development (by Gromov among others) of the theory of hyperbolic groups.