  Complex contact manifolds arise naturally in differential geometry, algebraic
geometry and exterior differential systems. Their classification would answer
an important question about holonomy groups. The geometry of such manifold $X$
is governed by the contact lines contained in $X$. These are related to the
notion of a variety of minimal rational tangents. In this review we discuss the
partial classification theorems of projective complex contact manifolds. Among
such manifolds one finds contact Fano manifolds (which include adjoint
varieties) and projectivised cotangent bundles. In the first case we also
discuss a distinguished contact cone structure, arising as the variety of
minimal rational tangents. We discuss the repercussion of the aforementioned
classification theorems for the geometry of quaternion-K\"ahler manifolds with
positive scalar curvature and for the geometry of second-order PDEs imposed on
hypersurfaces.
