Wallpaper groups are classifications of two-dimensional patterns that repeat in a regular way across a surface. There are 17 distinct groups, each defined by its symmetry properties, including translations, rotations, reflections, and glide reflections. These groups help in understanding how patterns can be arranged and repeated in art, architecture, and design. The study of wallpaper groups is rooted in group theory, a branch of mathematics that explores symmetry. Each group can be represented using a combination of geometric transformations, making it easier to analyze and categorize various patterns found in nature and human-made objects.