We put the known results on the antipode of a usual quasitriangular Hopf algebra into the framework of multiplier Hopf algebras. We illustrate with examples which cannot be obtained by using classical Hopf algebras. The focus of the present paper lies on the class of the so-called G-cograded multiplier Hopf algebras. By doing so, we bring the results of quasitriangular Hopf group-coalgebras (as introduced by Turaev) to the more general framework of multiplier Hopf algebras. (c) 2005 Elsevier Inc. All rights reserved.