Teichmüller theory is a branch of mathematics that studies the geometric structures of surfaces, particularly Riemann surfaces. It focuses on the concept of Teichmüller spaces, which represent different ways to deform a surface while preserving its complex structure. This theory helps mathematicians understand how surfaces can be classified and compared. One of the key aspects of Teichmüller theory is the Teichmüller metric, which provides a way to measure distances between different surfaces in the Teichmüller space. This metric allows for the exploration of the relationships between surfaces and has applications in various fields, including algebraic geometry and dynamics.