Flatland said:
The Universe does not have a center, meaning that it is either infinite or that if you travel in one direction long enough you will end back at the same point again.
That's a really clear way to put it. Concise too. Thanks! We're talking about the standard cosmic model(s)--the usual topics discussed here involve the models most cosmologists use and that you normally find in the literature.
That means matter distributed approximately UNIFORMLY throughout all existing space. No edge to the region occupied by matter. No edge to space itself--no "space outside of space".
These things would just add unnecessary complication to the model and there is no evidence for them so the Ockham simplicity criterion applies. One's always seeking the simplest model with the best fit to observation.
That's on a 3D level and well-understood by most of the people you see around here.
There's also a problem with the concept of "center" on a 4D level. You didn't address that in your post, so I'll try to say something (Brian Powell and Ben Crowell may have better ways of putting this or it may be covered in FAQ). I think the problem in that case is that there are different competing ideas.
In the old classic picture
there certainly is no center! speaking in a 4D sense because the model breaks down, blows up, is simply not defined at the start of expansion, which could extend over an infinite 3D hypersurface region according to the old classic picture.
So that corresponds to nobody's idea of "center" It is not even part of the manifold---the start of expansion is off the spacetime continuum on which the model is defined.
But there are different ideas of how to fix that, and the professional community has not yet CHOSEN a favorite.
So I feel as if I can't make any absolute unqualified statement about there being no "center". Most likely it won't be a POINT, in any case. But when some consensus does develop around one of the new models that resolve (get rid of) the singularity then maybe there WILL be something corresponding at least to a "central slice". Or maybe there won't be!
The central mathematical object might be a central infinite volume 3D slice or a central finite volume 3D hypersphere or torus---I don't think one can say---but there might be a central *something* in whatever consensus cosmic model emerges. Or might not! Who knows? Maybe some new mathematical object will be invented

something to replace the classical idea of a manifold (the classical space or spacetime continuum) and that will turn out to be the "center". Or there won't be a center in the new picture. I strongly believe we cannot foresee the future of math physics research---one of the nice things about it is the inherent unpredictability.
So one can say that in a 4D sense there is no center
so far. But there might be, when a consensus emerges around some model which includes the start of expansion in a nonsingular way.