In this paper, we investigate the existence and completeness of the wave operators W± = s‐limt→±∞ exp (itH) P exp (−itH0) corresponding to the quantum‐mechanical scattering of nonrelativistic particles by certain classes of impenetrable noncompact surfaces bounding domains Ω ⊆Rν (ν ⩾2) which contain a half‐space and are contained in another half‐space. Here, H0 is the usual negative (distributional) Laplacian−Δ in H0 = L2(Rν ), H is the negative Dirichlet Laplacian in H = L2(Ω), and P is an appropriate identification operator. Under these conditions, we prove by elementary methods that W± exist as partially isometric operators whose initial sets have a transparent physical meaning. Suppose now that the domain Ω ⊆Rν also has the periodicity property (x,xν)∊Ω→(x+l,xν)∊Ω when l ranges over a Bravais lattice in Rν−1 , where we write x∊Rν as (x,xν), with x∊Rν−1 and xν ∊R. Then (a) RanW± = Hscatt (H) and (b) W± are asymptotically complete, in the sense that H = Hscatt(H)⊕Hsurf(H). Here, Hscatt (H) and Hsurf (H) are suitably defined subspaces of scattering and surface states of H, respectively. Results (a) and (b) are proved by reducing the original scattering problem to a family of ’’scattering’’ problems in a periodicity cell of Ω, using direct‐integral methods, and by then using methods analogous to those of Lyford. The present work constitutes a rigorous foundation for the theory of scattering of low‐energy atomic beams by crystal surfaces, considered as impenetrable periodic barriers. Our methods should also be applicable to rigorous investigations of classical scattering by periodic surfaces with Dirichlet or Neumann boundary conditions.