  We discuss the possibility of constructing stable, static, spherically
symmetric, asymptotically flat Lorentzian wormhole solutions in General
Relativity coupled to a generalized Galileon field $\pi$. Assuming that
Minkowski space-time is obtained at $\partial \pi =0$, we find that there is
tension between the properties of the energy-momentum tensor required to
support a wormhole (violation of average null energy conditions) and stability
of the Galileon perturbations about the putative solution (absence of ghosts
and gradient instabilities). In 3-dimensional space-time, this tension is
strong enough to rule out wormholes with above properties. In higher
dimensions, including the most physically interesting case of 4-dimensional
space-time, wormholes, if any, must have fairly contrived shapes.
