This chapter considers the notion of a group in mathematics. It begins with a discussion of the problem of determining the symmetry of an object such as a planar shape, a higher-dimensional solid, a group, or an electric field. It then describes every group as a group of symmetries of some object and shows what it means for a group to be a group of symmetries of an object. These ideas are at the very heart of geometric group theory, the study of groups, spaces, and the interactions between them. The chapter also examines infinite groups, homomorphisms and normal subgroups, and group presentations. A number of exercises are included.