  Following an outline of Rezk, we give a construction of complex-analytic
$G$-equivariant elliptic cohomology for an arbitrary compact Lie group $G$ and
we prove some of its fundamental properties. The construction is parametrised
over the orbit category of the groupoid of principal $G$-bundles over
2-dimensional tori and generalises Grojnowski's construction of equivariant
elliptic cohomology.
