Spacetime Foam

The domain of quantum gravity is presumably bizarre by our usual standards. Since we do not have an adequate theory, we cannot make very precise statements, but qualitatively we have reason to believe that on this scale even space and time may become something other than our usual conceptions.
Of Wormholes and Such
For example, spacetime may develop "wormholes", such as the one illustrated in the adjacent figure for a 2-dimensional space. A wormhole could connect two regions of the Universe without going through the space in between the two points. (Although our diagram suggests that the distance through the wormhole is similar to the distance between the two mouths of the wormhole in the flat part of the space, the wormhole can be arranged such that the distance through it is very short relative to the distance between the mouths in the regular part of the space.)

Travel through Wormholes?

We are discussing spacetime at the microscopic level at the moment, but wormholes on a large scale are a favorite of science fiction writers since they suggest the possibility of essentially instantaneous travel between two different points of space by passing through the wormhole. However, general relativity indicates that even if such wormholes exist, any attempt to send a mass through it would collapse the wormhole unless it were stabilized by a material unlike any in our normal experience. This material would need to have the properties that we ascribed to the vacuum energy density in the discussion of the inflationary Universe and the the accelerated expansion of the Universe, and would effectively have to act like antigravity. Needless to say, at present we have no idea where to obtain such material in sufficient quantity to stabilize a wormhole (nor do we know where to find a wormhole to stabilize)!

Discontinuous Spacetime?
Even more disconcerting to our ordinary sensibilities about space and time is the possibility that the two are no longer even continuous below the Planck scale. This has been described rather poetically as the dissolution of the spacetime continuum into a frothing and bubbling "spacetime foam". Relativity implies that space and time are not what they seem, but with relativity we could at least retain the idea of spacetime as a continuous thing. With quantum gravity, even that may not be possible.
The Ultimate Free Lunch?
In quantum mechanics, even "empty space" (what physicists call the vacuum) is fluctuating with energy and particle-antiparticle pairs can materialize as excitations out the vacuum. The strangest of all the strange ideas associated with quantum gravity is that perhaps the Universe itself is a fluctuation in the "spacetime vacuum" (which corresponds to the absence of space and time). That is, perhaps at creation an expanding spacetime appeared out of "nothing" as a quantum fluctuation, giving birth to our Universe. This idea has been dubbed the "ultimate free lunch", since it corresponds to creating a Universe from literally nothing, not even space or time.