Basic modules of McLain groups are defined and investigated. These are a (possibly infinite dimensional) generalization of Andre’s basic modules of the multiplicative group of upper triangular square matrices over a finite field with 1’s on the main diagonal. The ring R need not be finite or commutative and modules of a McLain group are allowed to be infinite dimensional over an arbitrary field F. The set , totally ordered by , is allowed to be infinite. We show that distinct basic modules are disjoint, determine the dimension of the endomorphism algebra of a basic module, find when a basic module is irreducible, and we study the problem of finding a decomposition of a basic module as direct sum of irreducible submodules.