Conformal field theory is intimately connected to the theory of vertex algebras and the geometry of Riemann surfaces.

In this thesis a new algebro-geometric structure called global vertex algebra is defined on Riemann surfaces which is supposed to be a natural generalization of vertex algebras.

In order to define this structure a formal calculus of fields on Riemann surfaces is constructed. The basic objects in vertex algebra theory are fields. They are defined as formal Laurent series with possibly infinite principal part. The coefficients are endomorphisms.

As an example for such a structure the global vertex algebra of bosons of Krichever-Novikov type will be constructed.

At the beginning of this thesis the formal calculus of classical vertex algebras is introduced from the viewpoint of distributions in complex analysis.

Furthermore a graphical calculus for the computation of correlation functions of primary fields associated to affine Kac-Moody algebras is introduced.