An Abelian variety is a type of algebraic variety that is also a group, meaning it has a structure that allows for addition of its points. These varieties are defined over fields and are characterized by their complex torus structure, which means they can be represented as a quotient of a complex vector space by a discrete lattice. Abelian varieties play a crucial role in algebraic geometry and number theory. One of the key properties of Abelian varieties is that they are projective and complete, which means they can be studied using tools from both algebraic geometry and complex analysis. They are closely related to other mathematical concepts, such as Jacobian varieties, which arise in the study of algebraic curves. Abelian varieties also have applications in areas like cryptography and coding theory.