  Discrete conjugate systems are quadrilateral nets with all planar faces.
Discrete orthogonal systems are defined by the additional property of all faces
being concircular. Their geometric properties allow one to consider them as
proper discretization of conjugate, resp. orthogonal coordinate systems of
classical differential geometry. We develop techniques that allow us to extend
this known qualitative analogy to rigorous convergence results. In particular,
we prove the $C^\infty$-convergence of discrete conjugate/orthogonal coordinate
systems to smooth ones. We also show how to construct the approximating
discrete nets. Coordinate systems and their transformations are treated on an
equal footing, and the approximation results hold for transformations as well.
