Linear surface wave scattering theory is used to reconstruct the lateral heterogeneity under Europe and the Mediterranean using surface wave data recorded with the Network of Autonomously Recording Seismographs (NARS). The waveform inversion of the phase and the amplitude of the direct surface wave leads to a variance reduction of approximately 40% and results in phase velocity maps in the period ranges 30--40 s, 40--60 s and 60--100 s. A resolution analysis is performed in order to establish the lateral resolution of these inversions. Using the phase velocity perturbations of the three period bands, a two-layer model for the S velocity under Europe and the Mediterranean is constructed. The S velocity perturbations in the deepest layer (100--200 km) are much more pronounced than in the top layer (0--100 km), which confirms that the low-velocity zone exhibits pronounced lateral variations. In both layers the S velocity is low under the western Mediterranean, while the S velocity is high under the Scandinavian shield. In the deepest layer a high S velocity region extends from Greece under the Adriatic to northern Italy. Several interesting smaller features, such as the Massif Central, are reconstructed. One of the spectacular features of the reconstructed models is a sharp transition in the layer between 100 and 200 km near the Tomquist-Tesseyre zone. This would indicate that there is a sharp transition at depth between Central Europe and the East European platform. The waveform inversion of the surface wave coda leads to good waveform fits, but the reconstructed models are chaotic. This is due both to a lack of sufficient data for a good imaging of the surface wave energy on the heterogeneities and to an appreciable noise component in the surface wave coda. ¿ American Geophysical Union 1988