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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 realized 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 realized infinities. As an illustration we examine these issues in the case of the set of Friedmann–Lemaître–Robertson–Walker universes. Then we briefly summarize scenarios such as chaotic inflation, which suggest how ensembles of universe domains may be generated, and we 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. [ABSTRACT FROM AUTHOR]