From observations of swimming of the shell-less pteropod mollusc Clione antarctica we compare swimming velocities achieved by the organism using ciliated surfaces alone with velocities achieved by the same organism using a pair of flapping wings. Flapping dominates locomotion above a swimming Reynolds number ${\it Re}$ in the range 5–20. We test the hypothesis that ${\it Re} \approx$5–20 marks the onset of ‘flapping flight’ in these organisms. We consider the proposition that forward, reciprocal flapping flight is impossible for locomoting organisms whose motion is fully determined by a body length $L$ and a frequency $\omega$ below some finite critical value of the Reynolds number ${\it Re}_\omega = \omega L^2/\nu$. For a self-similar family of body shapes, the critical Reynolds number should depend only upon the geometry of the body and the cyclic movement used to locomote. We give evidence of such a critical Reynolds number in our data, and study the bifurcation in several simplified theoretical models. We argue further that this bifurcation marks the departure of natural locomotion from the low Reynolds number or Stokesian realm and its entry into the high Reynolds number or Eulerian realm. This occurs because the equilibrium swimming or flying speed $U_f$ obtained at the instability is determined by the mechanics of a viscous fluid at a value of ${\it Re}_f=U_f L/\nu$ that is not small.