We present ab-initio electronic structure results on the surface of 3× 3 adsorbates. In particular, we address the issue of metal-insulator instabilities, charge-density waves (CDWs) or spin-density waves (SDWs), driven by partly filled surface states and their 2D Fermi surface, and/or by the onset of magnetic instabilities. The focus is both on the newly discovered commensurate CDW transitions in the Pb/Ge(111) and Sn/Ge(111) structures, and on the puzzling semiconducting behavior of the Pb/Ge(111), K/Si(111):B and SiC(0001) surfaces. In all cases, the main factor driving the instability appears to be an extremely narrow surface state band. So far, we have carried out preliminary calculations for the Si/Si(111) surface, chosen as our model system, within the gradient corrected local density (LDA+GC) and local spin density (LSD+GC) approximations, with the aim of understanding the possible interplay between 2D Fermi surface and electron correlations in the surface+adsorbate system. Our spin-unrestricted results show that the 3× 3 paramagnetic surface is unstable towards a commensurate density wave with periodicity 3×3 and magnetization 1/3.