If all we can observe are things inside the observable universe, how do we know that anything even exists outside this boundary? I can see four ways of solving this problem.
1) We wait a while, the observable universe should get 'larger', so we should be able to observe more. I don't think this is practical though, since telescopes have only existed for a hundred years or so, whereas the age of the universe is many degrees larger. Also, galaxies are expanding outwards, so they will probably escape the observable universe horizon before we can observe them.
2) We go to another planet, and the 'centre' of the observable universe there is different, so in effect we can 'see' further then we could before. This is also impractical, we can't travel large enough distances for this effect to even have an inkling of an effect.
3) No way of knowing. All the matter that is within the observable universe could be all that there is, and we would have no way of knowing if anything even exists outside.
4) Theoretical calculations. Well scientists seem to be able to calculate (from redshifts?) the amount of dark matter, dark energy, and matter in the universe. That would mean that we would gain an understanding of what SHOULD be outside our observable universe (though we may never see it). This seems the most logical, but I have no idea how these calculations are carried out, any insight would be appreciated.