An Abelian surface is a type of complex algebraic variety that can be thought of as a two-dimensional generalization of an Abelian group. It is defined as a smooth projective variety that is also a group, meaning it has a group structure compatible with its geometric properties. Abelian surfaces can be represented as a product of two elliptic curves, which are one-dimensional analogs. These surfaces play a significant role in various areas of mathematics, including number theory, algebraic geometry, and cryptography. They are studied for their rich structure and connections to other mathematical objects, such as moduli spaces and automorphic forms.