Teichmüller spaces are mathematical structures that arise in the study of Riemann surfaces, which are one-dimensional complex manifolds. They provide a way to understand the different complex structures that can be placed on a given topological surface, capturing the idea of how these surfaces can be deformed. Each point in a Teichmüller space corresponds to a unique complex structure on a surface, while the space itself reflects the various ways these structures can be continuously transformed. This concept is essential in areas like algebraic geometry and hyperbolic geometry, where understanding the shape and form of surfaces is crucial.