We study the negating side of the Gromov-Oka Principle, namely the failure and limited validity of this important principle. The obstructions to the principle are based on the relation between conformal invariants of the source and the target. The conformal invariant used in this chapter is the collection of conformal modules of conjugacy classes of elements of the fundamental group of the manifold. For mappings from finite open Riemann surfaces to the twice punctured complex plane we will confirm Gromov’s prediction, that mappings from annuli into a complex manifold play a special role for understanding the homotopy problem of continuous mappings to holomorphic mappings. We give a complete description of all continuous mappings from any finite open oriented surface that are homotopic to holomorphic mappings for any orientation preserving complex structure with only thick ends.