The Universe that originated from the Big Bang is not infinite and its boundaries are based on cosmological inflation and the speed of light. It is not known (and it may not even be knowable) if there are other universes that inhabit our 4-D spacetime or multiverses for higher dimensions.

Evidence for the finite size and age of our Universe comes from three sources:

• The Hubble Constant: the redshift observed from distant galaxies showing they are receding from us--thus, if you reversed time, they came from a single point
• The Cosmic Microwave Background: broadly indicates a fairly homogenous universe and its very existence indicates a point where the universe cooled down to a point where atoms could form (hence, its radiation). It also supports inflation theory.
• The geological record coupled with solar nucleosynthesis observations of Population I through III stars--all of the elements that our universe is composed of are created in stars with the exception of hydrogen, helium, and lithium. The proportion of each material is dependent on star sizes and the gasses from which the stars form. If the Universe were infinite or existed for an infinitely long time, the values would equalize. They aren't. Some stars don't have many metals (their source material was closer to that of the Big Bang). Others have more (or have produced more). Some types of metals, like iron, can't be produced in small stars (like our own), so you know they weren't produced in fusion of that particular star.

And from a philosophical perspective:

*Olbers' paradox -- if the universe were infinite and infinity old, there would be light on every solid angle you could think of and thus we would be burned to a crisp.