As i understand it (i just do physics, not astronomy) it depends on what theory of the universe is correct
If it is infinite in all direction then there is no centre of the universe, and everything in the universe is simply moving away from each other, and not away from some arbitrary centre.
If it is a slab universe (inifinte in X and Y but not in Z) then there is a centre LINE through the universe, but not a centre point.
If it is a donut universe (think of a donut, where the good tasting stuff would be the universe) then there would be a centre of the universe, but that would be a centre of non existence, and it would not be possible to travel there, there would also be a centre curve around the middle of the torus ring, similar to that of the slab universe.
Oh and Big bang does have many problems with it, namely the singularity problem (alhtough if you consider the universe using complex maths you can make the singularity go away... but no one is exactly sure how to rectify thay) and also with the Matter Anti-Matter disparity.
There also does not seem to be uniformity in the background radiation of the universe, which also messes up Big Bang.
But like evolution, it is the best fit theory we have - but then there was a time when we thought Newtonian physics governed everything... then we looked at the very fast, very hot, and very small.