A crystal is a shape which tessellates the plane (or space) in a periodic way, so that the pattern repeats at regular intervals in all directions. The group of symmetries (translational, rotational, reflectional) of such a tessellation is called a crystallographic group. In two dimensions there are exactly 17 different kinds of symmetry, the so-called "wallpaper groups", which I'll describe. I'll also mention what happens in three dimensions, in hyperbolic space, and how you can make "quasiperiodic" tessellations (Penrose tilings) with five-fold symmetry.