For one-dimensional chains, transverse displacements of atoms produce zigzag structures, which are encountered in such diverse areas as polymers and metallic overlayers on semiconductor surfaces. Such displacements lead to an apparent reduction of translational symmetry but do not open a gap at the zone edge [gapless Peierls transition (GPT)]. The total energy remains analytic and hence the formation of zigzag chains on semiconductors is governed by material-specific interactions. The applications of the GPT ideas are illustrated by examining the stability issues and electronic structure of trans-(CH)x, Al chains on Si(001), and consequences of the tetragonal- orthorhombic transition in La2CuO4. © 1990 The American Physical Society.