Kac-Moody groups G over R have been conjectured to occur as
symmetry groups of supergravities in dimensions less than 3, and their integer
forms G(Z) are conjecturally U-duality groups. Mathematical
descriptions of G(Z), due to Tits, are functorial and not amenable
to computation or applications. We construct Kac-Moody groups over R
and Z using an analog of Chevalley's constructions in finite
dimensions and Garland's constructions in the affine case. We extend a
construction of Eisenstein series on finite dimensional semisimple algebraic
groups using representation theory, which appeared in the context of
superstring theory, to general Kac-Moody groups. This coincides with a
generalization of Garland's Eisenstein series on affine Kac-Moody groups to
general Kac-Moody groups and includes Eisenstein series on E10​ and
E11​. For finite dimensional groups, Eisenstein series encode the quantum
corrections in string theory and supergravity theories. Their Kac-Moody analogs
will likely also play an important part in string theory, though their roles
are not yet understood