  This is the first in a series of papers investigating the relationship
between the twisted equivariant K-theory of a compact Lie group G and the
"Verlinde ring" of its loop group. In this paper we set up the foundations of
twisted equivariant K-groups, and more generally twisted K-theory of groupoids.
We establish enough basic properties to make effective computations. Using the
Mayer-Vietoris spectral sequence we compute the twisted equivariant K-groups of
a compact connected Lie group G with torsion free fundamental group. We relate
this computation to the representation theory of the loop group at a level
related to the twisting.
