  This is the first comprehensive introduction to the authors' recent attempts
toward a better understanding of the global concepts behind spinor
representations of surfaces in 3-space. The important new aspect is a
quaternionic-valued function theory, whose "meromorphic functions" are
conformal maps into quaternions, which extends the classical complex function
theory on Riemann surfaces. The first results along these lines were presented
at the ICM 98 in Berlin. An important new invariant of the quaternionic
holomorphic theory is the Willmore energy. For quaternionic holomorphic curves
in the quaternionic projective line, i.e., the conformal 4-sphere, this energy
is the classical Willmore energy of conformal surfaces. Using these new
techniques, the article discusses the mean curvature sphere of conformal
surfaces in the 4-sphere, Willmore surfaces in the 4-sphere, Baecklund
tranformations of Willmore surfaces, super-conformal surfaces and twistor
projections, and a duality between Willmore surfaces in the 3-sphere and
minimal surfaces in hyperbolic 3-space. Finally, a new proof of Montiel's
recent classification of Willmore 2-spheres in the 4-sphere is given.
