  The paper develops the fundamentals of quaternionic holomorphic curve theory.
The holomorphic functions in this theory are conformal maps from a Riemann
surface into the 4-sphere, i.e., the quaternionic projective line. Basic
results such as the Riemann-Roch Theorem for quaternionic holomorphic vector
bundles, the Kodaira embedding and the Pluecker relations for linear systems
are proven. Interpretations of these results in terms of the differential
geometry of surfaces in 3- and 4-space are hinted at throughout the paper.
Applications to estimates of the Willmore functional on constant mean curvature
tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue
estimates on Riemannian spin bundles in dimension 2 are given.
