  A Clifford algebra model for M"obius geometry is presented. The notion of
Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced,
and the structure equations for adapted frames are derived. These equations are
discretized and the geometry of the occuring discrete nets and sphere
congruences is discussed in a conformal setting. This way, the notions of
``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal
systems'' are obtained --- the latter as a generalization of discrete
orthogonal systems in Euclidean space. The relation of a Cauchy problem for
discrete orthogonal nets and a permutability theorem for the Ribaucour
transformation of smooth orthogonal systems is discussed.
