Classical algebraic geometry uses algebraic techniques to study geometric objects defined as the vanishing sets of multivariate polynomials over algebraically closed fields. Algebraic geometry over groups is an analogous field of mathematics that arises from the study of equations over groups; the latter bears a lot of similarity to the former, and hence, its name. In this thesis, we begin by formalising the concept of an equation over a group. We then proceed to study group-theoretic counterparts to central notions of elementary algebraic geometry such as affine algebraic sets, coordinate rings, the Noetherian condition, the Zariski topology and the Nullestelensatz of David Hilbert. In the process of doing so, a theory of groups that parallels classical commutative algebra will be developed to aid our study of the theory of equations over groups.