  In this letter we address issues involved in quarkonium production near the
boundaries of phase space. It is shown that higher-order non-perturbative
contributions are enhanced in this kinematic region and lead to a breakdown of
the non-relativistic (NRQCD) expansion. This breakdown is a consequence of
sensitivity to the kinematics of soft gluon radiation and to the difference
between partonic and hadronic phase space. We show how these large corrections
can be resummed giving the dominant contribution to the cross section. The
resummation leads to the introduction of non-perturbative, universal
distribution functions. We discuss the importance of these shape functions for
several observables, in particular the energy distribution of photo-produced
$J/\psi$ close to the endpoint.
