Throughout recorded history, we attacked this crucial problem in three different ways: constructing religions, embarking in philosophical thought, and embracing the scientific method. If we leave religion and philosophy aside and only deal with scientific reasoning, we see two complementary attempts carried forth: the study of the cosmos, the "outside", and the study of the infinitely small, the "inside". It gives me a warm feeling to see how these two approaches have been found to be quite tightly intertwined as we progressed our understanding of fundamental science: we are on the right track, apparently.
But we are still quite far from our goal. While at some level we seem to understand how the universe works and how it has come together, we still struggle with some critical details: why is there more matter than antimatter ?, where is dark matter ? why is the universe accelerating its expansion ?, and so on.
Note that those above are hard but well-formulated questions: they focus on specific observed properties of our Universe. In contrast, we can only speculate wildly on the theme of what ignited the explosion, which leads us to ask ourselves what existed before the big bang: it is doubtful that this question has a meaning at all if we remain within the boundaries of what we call Science.
By addressing the question of what existed before the big bang we risk abandoning physics and entering one of the other two fields. But we can try and avert the risk by anchoring ourselves to basic principles of physics which we believe to be universal. Among them I think one can pick a very fundamental law: Whatever is not forbidden is compulsory !
The above is called the Totalitarian Law, and was first suggested by a novelist, T.H.White (in "The Once and Future King", Ace, 1996, p. 121), and then married and sponsored by Murray Gell-Mann, one of the fathers of the quark model. Although the law has been used in the context of quantum processes, it appears a very reasonable concept to elect as a basic principle.
Now, we have in front of us blatant evidence that the big bang, the event which gave rise to our Universe, is physically possible. It happened, 13.5 billion years ago, and it produced everything we observe around us, including ourselves. The evidence for the big bang is today so strong that it can be taken as a fact and a solid basis to construct hypotheses. So why should not we trust that, being a possible physical process, the big bang is repeatable ?
In my opinion, the justification of the hypothesis that the big bang is a unrepeatable process is tough. We do not need to know what mechanism ignited our Universe, but it seems to me that Ockham's razor forces us to prefer the simpler hypothesis that there is nothing "magical" at work, nothing that made the big bang a unique event which cannot occur again. Sure, a God could choose to create the Universe and then choose not to repeat His act; but in the absence of a God, whatever caused our Universe to come into existence should be there to do it again.
If the big bang is a repeatable process, it will repeat itself: not being impossible, it must happen again. Note that here I am ignoring the fact that time might be something quite different from what we perceive it to be: something with a beginning and an end just like spatial dimensions. One can argue that even if we associate the notion of time to the existence of our Universe, this still does not prevent us to imagining that the whole thing is repeatable.
So if we consider the big bang as a normal physical process, we are bound to accept the possibility that it will repeat itself an infinite number of times ! Not just two or three: an infinite number of times. And infinity is a very, very large number !
Of course, one should also allow for the possibility that an infinity of universes exists in parallel, occupying disjoint regions of spacetime. There is no appreciable difference in the conclusion: in both cases we are brought to consider the idea of an infinite collection.
What would it mean to accept the idea of an infinity of universes ? First of all we might imagine an eternal repetition of exactly the same universe, with the same value of physical constants and initial conditions, but such a determininstic outcome would be quite surprising in my opinion. The alternative is more plausible and maybe even more intriguing: we may consider that physical constants and initial conditions might be different every time.
Incidentally, here we are getting close to some of the ideas of string theory: string theory is an attempt at conceiving both particles and forces as different excitations of some basic entities called strings, the true elementary constituents of everything. Besides being already mindboggling from the outset, given that it "works" only if we assume that space-time has not four but a larger number of dimensions (e.g. 11 in M-theory), string theory wants us to accept as a working hypothesis to comprehend the universe around us a very upsetting fact: that our universe is just one in ten to the five hundredth (a HUGE number!) possible universes, which may arise due to as many different vacuum states of the theory. These different vacua correspond to different values for the physical constants on which the phenomenology of particles and forces is based, and therefore quite different universes.
Note that as an experimental physicist I do not consider string theory very attractive from the point of view of a candidate theory of everything, because it negates our chances to verify its predictions: the "landscape" of possible realizations of these string vacua makes it impossible to falsify string theory as a scientific theory. As speculative material, however, it may fit in the picture of an infinity of universes, each one different from all others not just in the initial conditions but also in the actual physical laws it obeys.
A totally different idea is also connected to what we are discussing: the "many-worlds" interpretation of quantum mechanics by Hugh Everett. This is an attempt to make sense of the wave-function of quantum systems by hypothesizing that any possible realization of it is real: a infinite number of parallel universes exists, and quantum decoherence allows us to experience only one of them.
Anyway, let's leave these fancy theoretical ideas aside and consider what we have come to hypothesize: the infinity of the number of universes that exist or that will come to existence guarantees that another universe completely undistinguishable from ours will arise at some point. Wait - not just at some point: it will arise an infinite number of times !
Suddenly we come to terms with what this implies for each of us. In such a picture we are born, acquire self-awareness, live, and die. As we die we cease to exist, but if we are materialistic enough to accept that what our self-awareness is is the collection and interrelation of our neural circuits, synapses, activation potentials, and that these are bound to some day come again to existence in the very same form, we may think of this as a resurrection. And an instantaneous one for our conscience -however unaware it will be of its past instantiations: the unimaginable amount of time it has passed since the same biological configuration came to existence is immaterial from the personal point of view of an individual. We die, and we are reborn; no matter how much time that "and" corresponds to.
The above is as close as I personally can get to constructing a belief which in some way allows me to accept the idea of my mortality. It requires one to make several assumptions, but I believe it does not blatantly violate any physical law.
If we take this stand we are not as lucky as the lucky few that get their resurrected in a perfect computer emulation just before the big crunch in Frank Tipler's "The Physics of Immortality": they do live forever in a sort of heaven, while we still "grunt and sweat under a weary life" (you win a sucked mint if you recognize this quote); and we are bound to do so an infinite number of times!
On a second thought, I fail to cheer too much at this picture. Sure, if we are bound to live again we will one day meet again people we have lost, etcetera, etcetera. However, I am disturbed by one thing. We must conceive the coming to existence of any -really any- different possibilities, in addition to those exactly equal to those we are living now. Just as in the many-worlds interpretation of quantum mechanics, all possible realizations of physical reality must come to existence, if we have to accept the totalitarian principle. This is quite upsetting if you think about it seriously!