One-dimensional (1D) properties with strong electron correlation and various instabilities have been found recently in metallic chains or ribbons generated by metal adsorption of sub-monolayers on insulating substrates like Si(111) or Si(557). The unavoidable coupling to 2D and 3D for all these cases results in a wealth of new phenomena. Here we concentrate on the Pb/Si(557) system, where these couplings in 2D and 3D become very obvious, and discuss structural self-stabilization as the origin of 1D electronic transport. Finally, a new 1D state generated by step decoration is described and its properties are discussed.