For generic non-integrable systems we show that a semiclassical prediction of tunnelling rates between regular and chaotic phase-space regions is possible. Our prediction is based on complex paths which can be constructed despite the obstacle of natural boundaries. The semiclassically obtained tunnelling rates are in excellent agreement with numerical tunnelling rates for the standard map where few complex paths dominate. This gives a semiclassical foundation of the long-conjectured and often-observed exponential scaling with Planck's constant of regular-to-chaotic tunnelling rates.