  An n-dimensional submanifold X of a projective space P^N (C) is called
tangentially degenerate if the rank of its Gauss mapping \gamma: X ---> G (n,
N) satisfies 0 < rank \gamma < n.
  The authors systematically study the geometry of tangentially degenerate
submanifolds of a projective space $P^N (\mathbf{C})$. By means of the focal
images, three basic types of submanifolds are discovered: cones, tangentially
degenerate hypersurfaces, and torsal submanifolds. Moreover, for tangentially
degenerate submanifolds, a structural theorem is proven. By this theorem,
tangentially degenerate submanifolds that do not belong to one of the basic
types are foliated into submanifolds of basic types. In the proof the authors
introduce irreducible, reducible, and completely reducible tangentially
degenerate submanifolds. It is found that cones and tangentially degenerate
hypersurfaces are irreducible, and torsal submanifolds are completely reducible
while all other tangentially degenerate submanifolds not belonging to basic
types are reducible.
