We discuss various approaches to constructing analogs of  Lie groups for infinite dimensional Lie algebras. We are particularly interested in the Monster Lie algebra m, a generalized Kac--Moody algebra. We describe the construction of a group G(m), associated to m, given by generators and relations, as well as a group U^+ of automorphisms of a completion m^ of m. We also describe a variation of these constructions that is related to the action of the Monster finite simple group on m. The talk will be based on various joint works with D. Addabbo, E. Jurisich, M. Khaqan and Scott H. Murray.