Two graphs are minor-equivalent if each is isomorphic to a minor of the other. In this paper, we give structural characterizations of the minor equivalence classes of the infinite full grid GZ×L and of the infinite half-grid GZ×N. A corollary of these results states that every minor of GZ×z that has a minor isomorphic to GZ×N is minor-equivalent to one of GZ×Z or GZ×N. © 1999 Elsevier Science B.V. All rights reserved