Diophantine Geometry is a branch of mathematics that studies solutions to polynomial equations with integer or rational number solutions. It combines techniques from algebraic geometry and number theory to explore the properties of these equations and their geometric representations. The field is named after the ancient mathematician Diophantus, who is known for his work on equations that seek integer solutions. Researchers in Diophantine Geometry investigate various types of equations, such as elliptic curves and hyperbolic curves, to understand their structure and the nature of their solutions.