One of the foundation stones of physics is the second law of thermodynamics, which says, in effect, that things get more “random” as time progresses. Accordingly, the beginning of the universe-the “big bang”-must have been an extraordinarily precisely organized (i.e. very non-random) state. But the particular nature of this state provides us with an apparent paradox, raising the profound issue of how such a special state can have come about. In this talk, a novel, perhaps outrageous, proposal is suggested, which involves an examination of what is to be expected, also, in the very remote future of our universe, with its observed accelerated expansion. It has relevance, also, to some other deep puzzles of physics, from the quantum to the cosmological scale.