At incident positron energies below the threshold for positronium atom formation, there are many cases in which annihilation rates for molecules are far in excess of that possible on the basis of simple two-body collisions. We now understand that this phenomenon is due to positron attachment to molecules mediated by vibrational Feshbach resonances. The attachment enhances greatly the overlap of the positron with molecular electrons and hence increases the probability of annihilation. Furthermore, measurements of the annihilation spectra as a function of incident positron energy provide a means of measuring positron-molecule binding energies. In this paper we present an overview of our current understanding of this process, highlighting key results and discussing outstanding issues that remain to be explained.