All available observational (and theoretical) evidence is consistent with the 'Big Bang Model', namely, that space and time -- the Universe -- began a finite time ago in the past, and that spacetime -- the fundamental, underlying structure of the Universe -- has been expanding ever since. Galaxies and clusters of galaxies are apparently moving away from one another as they are carried along in the expanding framework of space and time.
The apparent recession of the galaxies is described by Hubble's law of redshifts, and is the same in all directions (i.e. it is isotropic). The velocity of expansion increases linearly (approximately) with distance (v = H.d). Except for 'local' details, the Universe appears to be uniform as well as isotropic. This is expressed by the 'Cosmological Principle': the Universe looks the same for observers everywhere. There is no preferred/special location in space. All locations in the Universe are equivalent.
Looking out into space is equivalent to looking back in time. In order to 'see' the Big Bang event I must look as far out in space, or as far back in time, as possible. The observable limit of our Universe is a distance at which the velocity of recession equals the speed of light. That observational limit exists in all directions. It is, in effect, a spherical surface centred on us. In a sense, then, the 'centre' of the Universe coincides with its 'edge'. But all locations are equivalent. An hypothetical observer located at our observable limit would detect us at her observable limit. But her observable limit -- the edge of her observable Universe -- would be equated to the 'centre'. Hence, every location in space -- including our location -- is at the edge/centre/observable limit of other observers. Therefore, the centre of the Universe is everywhere. The Universe should not be thought of as something akin to a balloon which is expanding outward into empty space. All of space and time are within our Universe. There is no 'outside'. There was no 'before'. The Universe -- space and time -- is unfolding, and all locations in space and time are partaking in the expansion. (There are several fundamentally important and difficult questions in cosmology for which answers are lacking. Those pertain to such things as the dark, or hidden, matter, cosmic inflation, and the discovery early in 1998 that the rate of expansion appears to be accelerating. Those questions do not alter the general picture which I have described above.)