Strictly speaking, a wallpaper pattern lives in the plane. When we attempt to weave a wallpaper pattern from bands, as in the image shown, the result is no longer a planar object. The symmetry group of this pattern is one of the eighty layer groups. The main point of this presentation is to familiarize the audience with these less well-known groups and highlight their importance in the study of symmetry. For many of the wallpaper groups, there are two ways to extend the action on the plane to an action on space that preserve the base plane. For instance, in the example, the mirror symmetry in the wallpaper group becomes a flip symmetry of the woven pattern in space. This leads to a rich set of examples, which we find beautiful as well as mathematically interesting.