Over the past few years, evidence has begun to accumulate suggesting that spacetime may undergo a "spontaneous dimensional reduction" to two dimensions near the Planck scale. I review some of this evidence, and discuss the (still very speculative) proposal that the underlying mechanism may be related to short-distance focusing of light rays by quantum fluctuations.

To appear in Proc. of the Sixth International School on Field Theory and Gravity, Petropolis, Brazil, 2012