Results of computer simulations of diffusion and relaxation in the liquid and the glassy state are presented for model systems and Cu33Zr67 and Se. The diffusion constant and the intermediate self scattering function show the typical behavior upon quenching from the liquid to the glass. The diffusion constant in the undercooled liquid can be fitted both with a Vogel-Fulcher or a mode coupling law. However, its pressure derivative, the activation volume, clearly follows the prediction of the mode coupling theory. From the diffusional isotope effect in the liquid we conclude that collectivity increases upon quenching to the glass transition. Checking the relaxations in the glass we find collective hopping of chains of atoms as elementary process. Both in the glassy state and in the undercooled liquid we observe an increase oc ∝√ι ofthe non-Gaussianity of the self correlation function, which measures the dynamic heterogeneity. This can be understood from the collective jump mechanism.