We demonstrate "bendotaxis," a novel mechanism for droplet self-transport at small scales. A combination of bending and capillarity in a thin channel causes a pressure gradient that, in turn, results in the spontaneous movement of a liquid droplet. Surprisingly, the direction of this motion is always the same, regardless of the wettability of the channel. We use a combination of experiments at a macroscopic scale and a simple mathematical model to study this motion, focusing in particular on the timescale associated with the motion. We suggest that bendotaxis may be a useful means of transporting droplets in technological applications, e.g., in developing self-cleaning surfaces, and discuss the implications of our results for such applications.