Class Field Theory is one of the major achievements in number theory of the first half of the 20h century. Among other things, Artin reciprocity showed that the set of all unramified extensions of a number or function field can be described by an abelian object only depending on intrinsic data of the field.

In the language of Grothendieck´s algebraic geometry, the theorems of class field theory can now be reformulated as a theorem about certain one- dimensional "arithmetic" schemes, which correspond precisely to number fields and regular function fields. A natural question is now to ask for a generalisation of class field theory to arithmetic schemes of higher dimension. The goal of this talk is to introduce a generalisation of the class group to higher dimensional schemes which only depends on the 0- and 1- dimensional data contained in the scheme. This will allow us to construct the reciprocity homomorphism from the (abelian) class group to the abelianised fundamental group. A key idea to showing that this is has the required properties for a reciprocity homomorphism is the one-to-one correspondence of subgroups on both sides.

In this first talk, I shall begin by presenting the Wiesend class group C_X of a scheme X (an abelian group), construct the reciprocity homormphism and introduce the non-abelian concept of a covering datum.