Teichmüller Theory is a branch of mathematics that studies the geometric structures of surfaces, particularly Riemann surfaces. It focuses on understanding how these surfaces can be deformed while preserving certain properties. This theory provides a way to classify surfaces based on their shapes and complex structures. One of the key concepts in Teichmüller Theory is the Teichmüller space, which represents all possible complex structures on a given surface. This space helps mathematicians analyze the relationships between different surfaces and their geometric properties, leading to insights in areas like algebraic geometry and dynamics.