  Kac-Moody groups over finite fields are finitely generated groups. Most of
them can naturally be viewed as irreducible lattices in products of two closed
automorphism groups of non-positively curved twinned buildings: those are the
most important (but not the only) examples of twin building lattices. We prove
that these lattices are simple if and only if the corresponding buildings are
(irreducible and) not of affine type (i.e. they are not Bruhat-Tits buildings).
In fact, many of them are finitely presented and enjoy property (T). Our
arguments explain geometrically why simplicity fails to hold only for affine
Kac-Moody groups. Moreover we prove that a nontrivial continuous homomorphism
from a completed Kac-Moody group is always proper. We also show that Kac-Moody
lattices fulfill conditions implying strong superrigidity properties for
isometric actions on non-positively curved metric spaces. Most results apply to
the general class of twin building lattices.
