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The idea of a multiverse -- an ensemble of universes or universe domains -- has received increasing attention in cosmology, both as the outcome of the originating process that generated our own universe, and as an explanation for why our universe appears to be fine-tuned for life and consciousness. Here we carefully consider how multiverses should be defined, stressing the distinction between the collection of all possible universes, and ensembles of really existing universes, which are essential for an anthropic argument. We show that such realised multiverses are by no means unique, and in general require the existence of a well-defined and physically motivated distribution function on the space of all possible universes. Furthermore, a proper measure on these spaces is also needed, so that probabilities can be calculated. We then discuss several other major physical and philosophical problems which arise in the context of ensembles of universes, including the emergence and causal effectiveness of self-consciousness, realized infinities, and fine- tuning, or the apparent need for very special initial conditions for our universe -- whether they or generalized generic primordial conditions are more fundamental. Then we briefly summarise scenarios like chaotic inflation, which suggest how ensembles of universe domains may be generated, and point out that the regularities which must underlie any systematic description of truly disjoint multiverses must imply some kind of common generating mechanism. Finally, we discuss the issue of testability, which underlies the question of whether multiverse proposals are really scientific propositions., Comment: 37 pages

The idea of a multiverse -- an ensemble of universes -- has received increasing attention in cosmology, both as the outcome of the originating process that generated our own universe, and as an explanation for why our universe appears to be fine-tuned for life and consciousness. Here we carefully consider how multiverses should be defined, stressing the distinction between the collection of all possible universes, and ensembles of really existing universes that are essential for an anthropic argument. We show that such realised multiverses are by no means unique. A proper measure on the space of all really existing universes or universe domains is needed, so that probabilities can be calculated, and major problems arise in terms of realised infinities. As an illustration we examine these issues in the case of the set of Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) universes. Then we briefly summarise scenarios like chaotic inflation, which suggest how ensembles of universe domains may be generated, and point out that the regularities which must underlie any systematic description of truly disjoint multiverses must imply some kind of common generating mechanism. Finally, we discuss the issue of testability, which underlies the question of whether multiverse proposals are really scientific propositions., Comment: 32 pages, 2 figures Completely revised text addressing physics issues more fully and omitting philosophical discussions previously included