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Van der Waals-Casimir dispersion interactions between two apposed graphene layers, a graphene layer and a substrate, and in a multilamellar graphene system are analyzed within the framework of the Lifshitz theory. This formulation hinges on a known form of the dielectric response function of an undoped or doped graphene sheet, assumed to be of a random-phase-approximation form. In the geometry of two apposed layers, the separation dependence of the van der Waals-Casimir interaction for both types of graphene sheets is determined and critically compared with some well-known limiting cases. In a multilamellar array, the many-body effects are quantified and shown to increase the magnitude of the van der Waals-Casimir interactions.