In the classical theory of surfaces, there is a remarkable and subtle transformation discovered by Lie, Bianchi, and B\"acklund which converts any surface of constant negative curvature into another. The PDE governing such surfaces is nonlinear and difficult to work with, yet the transformation of one solution into another only involves the integration of some auxiliary ODEs. The existence of such a transformation points to the total integrability of an underlying PDE. I will develop computational tools (geometric exterior differential systems) which are closely adapted to the geometry of such problems. With these tools, we can automatically discover and derive the classical B\"acklund transformation, along with novel B\"acklund transformations of special surfaces in arbitrary geometries.