Do wormholes really exist?
General relativity asserts that certain configurations of spacetime, called wormholes, will satisfy the equations that govern spacetime and this is why they are part of mainstream physics. However, there are two qualifying observations:
If at some time there are no wormholes anywhere, then it seems that none can form by any physical process of a type which could in principle be described by classical as opposed to quantum physics. I say "seems" here because this has not been proved in complete generality but it is certainly true in all configurations that one can reasonably expect to come about by natural processes.
The mere existence of a wormhole does not necessarily imply that travel through the wormhole is possible, because of stability considerations (and some kinds of wormhole are not timelike anyway). The stability problem is of two kinds: the impact on the traveler of other stuff such as light passing into the wormhole, and the fact that their very presence (their mass) may cause the spacetime to distort in such a way as to seal the wormhole.
At the moment it is hard, therefore, to assert with any confidence that there really could be a traversable wormhole somewhere.
However one should add that when we bring in quantum physics then the range of possibilities becomes a lot richer. It seems now that one ought not to discount the possibility of wormholes, at least at microscopic scales, which is another way of saying that the structure of spacetime may be very complicated at the Planck scale. However this is not the kind of thing that ordinary people mean when they think about wormholes in science fiction scenarios.
No, as far as we know there are no wormholes anywhere in the universe. In fact, we know of no realistic physical mechanism that would lead to the creation of a wormhole in nature.
Worm holes are the product of the fact that the differential equations for general relativity have (permit) closed loop solutions. This implies that at the very MOST, the very simple and neat equations allow, looped space and time. These things need a real interpretation which isn't easy. One is that, space gets so warped that it can loop as opposed to being flat, has to do with geometry, and Klein bottle type stuff. But I suppose this leads to unintuitive phenomena, like travelling one way, and ending up a larger distance away than what would seem possible via normal paths.
Similarly with time, I suppose, which is even harder to apply to reality, but would be forward temporal destination leading to past times. Some of these are paradoxical and some reject they could be possible.
On normal scales there is no way to get feedback to the issue. On small scales there are issues with difficulty in making observation again. So I would presume there is little actual evidence in either direction. Only correlation with common measurable phenomena with general relativity field equations. There is no reason necessarily to presume they follow such curves far beyond these regions. It could be that different phenomena emerge which if predictable require new formulations, which do not have such closed solutions.
It is interesting to note that NASA does research a warp drive. This drive would utilize warping space time, so that faster than light travel is achieved. But no holes are utilized, only the contraction of space. This requires the contraction of space, which brings the object closer then normal, allowing faster transit than what would normally be possible even at light speed travel. This method is much more realizable but not to a level that is actually useful for transporting anything yet. But no puncturing of space and time travel is done. No observable puncture has ever been discovered. Black holes could be an example if General Relativity stands, but if not it could be just an extremely massive planet/star thing.
If we think, to any one that did not know better, and thought the earth was flat, because everything locally is flat, it would be similarly odd, to see someone leave travelling one direction, and arrive from the other. So it is presumable that spatial punctures could exist, as well as finite closed space. Closed time seems a little more problematic. But I do not know of any evidence for punctured space. And it can not be known if such things which are solutions to the equations, follow reality.