In this paper we generalize the notion of Hopf algebra. We consider an algebra A, with or without identity, and a homomorphism Δ \Delta from A to the multiplier algebra M ( A ⊗ A ) M(A \otimes A) of A ⊗ A A \otimes A . We impose certain conditions on Δ \Delta (such as coassociativity). Then we call the pair ( A , Δ ) (A,\Delta ) a multiplier Hopf algebra. The motivating example is the case where A is the algebra of complex, finitely supported functions on a group G and where ( Δ f ) ( s , t ) = f ( s t ) (\Delta f)(s,t) = f(st) with s , t ∈ G s,t \in G and f ∈ A f \in A . We prove the existence of a counit and an antipode. If A has an identity, we have a usual Hopf algebra. We also consider the case where A is a ∗ \ast -algebra. Then we show that (a large enough) subspace of the dual space can also be made into a ∗ \ast -algebra.