  We survey several results in value distribution theory for parabolic Riemann
surfaces. Let Y be a parabolic Riemann surface, i.e. subharmonic functions
defined on Y are constant. We discuss Nevanlinna's theory for holomorphic maps
f from Y to the projective line. The results we obtain parallel the classical
case Y is the complex line, as we describe now.
  Let X be a manifold of general type, and let A be an ample line bundle on X.
It is known that there exists a holomorphic jet differential P (of order k)
with values in the dual of A. If the map f has infinite area and if Y has
finite Euler characteristic, then we show that f satisfies the differential
relation induced by P.
  As a consequence, we obtain a generalization of Bloch Theorem concerning the
Zariski closure of maps f with values in a complex torus. An interesting
corollary of these techniques is a refined Ax-Lindemann theorem, for which we
give a quick proof.
  We then study the degree of Nevanlinna's current T[f] associated to a
parabolic leaf of a foliation F by Riemann surfaces on a compact complex
manifold. We show that the degree of T[f] on the tangent bundle of the
foliation is bounded from below in terms of the counting function of f with
respect to the singularities of F, and the Euler characteristic of Y. In the
case of complex surfaces of general type, we obtain a complete analogue of
McQuillan's result: a parabolic curve of infinite area and finite Euler
characteristic tangent to F is not Zariski dense.
