  We expose the elliptic quantum groups in the Drinfeld realization associated
with both the affine Lie algebra \g and the toroidal algebra \g_tor. There the
level-0 and level \not=0 representations appear in a unified way so that one
can define the vertex operators as intertwining operators of them. The vertex
operators are key for many applications such as a derivation of the elliptic
weight functions, integral solutions of the (elliptic) q-KZ equation and a
formulation of algebraic analysis of the elliptic solvable lattice models.
Identifying the elliptic weight functions with the elliptic stable envelopes we
make a correspondence between the level-0 representation of the elliptic
quantum group and the equivariant elliptic cohomology. We also emphasize a
characterization of the elliptic quantum groups as $q$-deformations of the
W-algebras.
