  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.
