Kac-Moody groups are `infinite dimensional' analogs of Lie groups associated to Kac-Moody algebras. When R is a commutative ring, for any symmetrizable generalized Cartan matrix A, the Kac-Moody group G_A( R ) is the value of the Tits' functor G_A over R. This abstract definition of Tits is axiomatic in nature and not amenable to computation or to applications. We construct Kac--Moody groups over arbitrary fields and the integers Z using an analog of Chevalley's constructions in finite dimensions and Garland's constructions in the affine case. We give a finite presentation for an important class of Kac-Moody groups over Z, including E_{10}(Z). The motivation for this work comes directly from high energy theoretical physics and answers questions relating to mathematical structures in models for supergravity theories.