We discuss the electronic transport through molecules in the Kondo regime. We concentrate here on the influence of molecular vibrations. Two types of vibrations are investigated: (i) the breathing internal molecular modes, where the coupling occurs between the molecular deformation and the charge density, and (ii) the oscillations of molecule between the contacts, where the displacement affects the tunneling. The system is described by models which are solved numerically using Schoenhammer-Gunnarsson's projection operators and Wilson's numerical renormalization group methods. Case (i) is considered within the Anderson-Holstein model. Here the influence of the phonons is mainly to suppress the repulsion between the electrons at the molecular orbital. Case (ii) is described by a two-channel Anderson model with phonon-assisted hybridization. In both cases, the coupling to electrons softens the vibrational mode and in the strong coupling regime makes the displacement unstable to perturbations that break the symmetry of the confining potential. For instance, in case (ii) when the frequency of oscillations decreases below the magnitude of perturbation breaking the left-right symmetry, the molecule will be abruptly attracted to one of the electrodes. In this regime, the Kondo temperature increases but the conductance through the molecule is suppressed.