Transport through single molecules differs from transport through more conventional nanostructures such as quantum dots by the coupling to few well-defined vibrational modes. A well-known consequence of this coupling is the appearance of vibrational side bands in the current-voltage characteristics. We have recently shown that the coupling to vibrational modes can lead to new quantum transport effects for two reasons. (i) When vibrational equilibration rates are sufficiently slow, the transport current can drive the molecular vibrations far out of thermal equilibrium. In this regime, we predict for strong electron-phonon coupling that electrons pass the molecule in avalanches of large numbers of electrons. These avalanches consist themselves of smaller avalanches, interrupted by long waiting times, and so on. This self-similar avalanche transport is reflected in exceptionally large current (shot) noise, as measured by the Fano factor, as well as a power-law frequency spectrum of the noise. (ii) Due to polaronic energy shifts, the effective charging energy of molecules may be strongly reduced compared to the pure Coulomb charging energy. In fact, for certain molecules the effective charging energy U can become negative, a phenomenon known in chemistry as potential inversion. We predict that transport through such negative-U molecules near charge-degeneracy points, where the Coulomb blockade is lifted, is dominated by tunnelling of electron pairs. We show that the dependence of the corresponding Coulomb-blockade peaks on temperature and bias voltage is characteristic of the reduced phase space for pair tunnelling.