Let G be any discrete group. Consider the algebra A of all complex functions with finite support on G with pointwise operations. The multiplication on G induces a comultiplication Delta on A by (Delta f)(p, q) = f(pq) whenever f is an element of A and p, q is an element of G. If G is finite, one can identify the algebra of complex functions on G x G with A x A so that Delta: A --> A x A. Then (A, Delta) is a Hopf algebra. If G is infinite, we still have Delta(f)(g x 1) and Delta(f)(1 x g) in A x A for all f and g. In this case (A, Delta) is a multiplier Hopf algebra. In fact, it is a multiplier Hopf *-algebra when A is given the natural involution defined by f*(p) = <(f/(p))over bar> for all f is an element of A and p is an element of G. In this paper we call a multiplier Hopf *-algebra (A, Delta) a discrete quantum group if the underlying *-algebra A is a direct sum of full matrix algebras. We study these discrete quantum groups and we give a simple proof of the existence and uniqueness of a left and a right invariant Haar measure. (C) 1996 Academic Press, Inc.