Results from time-dependent two-dimensional hydrodynamic simulations of the global heliosphere suggest that drag between the plasma ions and the interstellar neutrals, caused by charge-exchange collisions, may cause the heliopause to be hydrodynamically unstable. Both ions and neutrals are treated as fluids coupled by charge-exchange collisions. The neutral-ion drag is proportional to the plasma density and introduces an effective gravity in the direction of the neutral flow, which, because the interstellar plasma is much denser than the heliosheath plasma, causes a Rayleigh-Taylor-like instability to develop. The heliopause is unstable only near the stagnation point at the ''nose'' of the heliosphere. In the simulations, the heliopause is seen to oscillate nonlinearly about its equilibrium position with a timescale of the order of a hundred years and amplitudes of tens of AUs. Growth rates from the simulations are in reasonable agreement with theoretical estimates. The possible stabilizing influence of energetic solar wind neutrals, neglected in the present model, is discussed. Implications of this instability on the interpretation of the Voyager 2--3 kHz emissions are also discussed. ¿ American Geophysical Union 1996