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We study the dynamics of the restricted planar three-body problem near meanmotion resonances, i.e. a resonance involving the Keplerian periods of the twolighter bodies revolving around the most massive one. This problem is oftenused to model Sun--Jupiter--asteroid systems. For the primaries (Sun andJupiter), we pick a realistic mass ratio $\mu=10^{-3}$ and a small eccentricity$e_0>0$. The main result is a construction of a variety of non local diffusingorbits which show a drastic change of the osculating (instant) eccentricity ofthe asteroid, while the osculating semi major axis is kept almost constant. Theproof relies on the careful analysis of the circular problem, which has ahyperbolic structure, but for which diffusion is prevented by KAM tori. Weverify certain non-degeneracy conditions numerically.  Based on the work of Treschev, it is natural to conjecture that diffusiontime for this problem is $\sim \frac{-\ln (\mu e_0)}{\mu^{3/2} e_0}$. We expectour instability mechanism to apply to realistic values of $e_0$ and we giveheuristic arguments in its favor. If so, the applicability of Nekhoroshevtheory to the three-body problem as well as the long time stability becomequestionable.  It is well known that, in the Asteroid Belt, located between the orbits ofMars and Jupiter, the distribution of asteroids has the so-called Kirkwood gapsexactly at mean motion resonances of low order. Our mechanism gives a possibleexplanation of their existence. To relate the existence of Kirkwood gaps withArnold diffusion, we state a conjecture on its existence for a typical$\eps$-perturbation of the product of a pendulum and a rotator. Namely, wepredict that a positive conditional measure of initial conditions concentratedin the main resonance exhibits Arnold diffusion on time scales $\frac{- \ln\eps}{\eps^{2}}$.

Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem