There is strong evidence for a minimal length in nature: the Planck length L = 10^{-33} cm. On this scale, quantum fluctuations of the metric are large, and the meaning of spacetime breaks down.
Recently, I and two collaborators at Caltech showed that no device (not even a gedanken experiment) is capable of measuring a distance less than the Planck length. (The paper is published in Physical Review Letters.) By "measuring a distance less than the Planck length" we mean, technically, resolve the eigenvalues of the position operator to within L. (Previous work on this problem had not been very careful in defining minimum length, and to obtain a clean result we had to be a bit careful.) The only assumptions in our argument are the uncertainty principle from quantum mechanics and a dynamical criterion for gravitational collapse from classical general relativity called the hoop conjecture.
An implication of the result is that there may only be a finite number of degrees of freedom per unit volume in our universe - no true continuum of space or time. This means that there is only a finite amount of information or entropy in our universe (or at least in any finite patch of it).
One of the main problems encountered in the quantization of gravity is a proliferation of divergences coming from short distance fluctuations of the metric (or graviton). However, these divergences might only be artifacts of perturbation theory: minimum length, which is itself a non-perturbative effect, might provide a cutoff which removes the infinities. This conjecture could be verified by lattice simulations of quantum gravity (for example, in the Euclidean path integral formulation), by checking to see if they yield finite results even in the continuum limit.
It certainly would be interesting if the solution to the divergences in quantum gravity turned out to be something as simple as a length cutoff. According to Sundrum, a graviton of finite size might also address the Cosmological Constant problems.
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