Surfaces of insulating single crystals (like Si at and below room temperature) provide perfect templates for ordered arrays of atomic wires, which in many cases exhibit one-dimensional (1D) properties. The physics in these quasi-one-dimensional systems is quite intriguing and gives rise to the observation of new phenomena such as spin and charge order. Here we present a short overview, concentrating on Pb on Si(557) close to monolayer coverage, which a priori is an anisotropic 2D system, as shown, e.g., by magnetoconductance or by plasmonic excitations. At certain critical conditions, however, it turns into a 1D system, as most obvious from dc conductance and angular resolved photoemission. Conditions and stability of its occurrence will be discussed.