For an action a of a groupGon an algebraR(over C), the crossed productRxαGis the vector space ofR-valued functions with finite support inG, together with the twisted convolution product given bywherep∈G. This construction has been extended to the theory of Hopf algebras. Given an action of a Hopf algebraAon an algebraR, it is possible to make the tensor productR⊗Ainto an algebra by using a twisted product, involving the action. In this case, the algebra is called the smash product and denoted by R#A. In the group case, the action a ofGonRyields an action of the group algebra CGas a Hopf algebra onRand the crossedRxαGcoincides with the smash productR#CG. In this paper we extend the theory of actions of Hopf algebras to actions of multiplier Hopf algebras. We also construct the smash product and we obtain results very similar as in the original situation for Hopf algebras.