Conformal geometry is a branch of mathematics that studies the properties of shapes that are preserved under angle-preserving transformations. These transformations, known as conformal mappings, allow for the comparison of geometric structures while maintaining the angles between curves. This field is particularly useful in understanding complex shapes and surfaces. One of the key concepts in conformal geometry is the Riemann surface, which provides a way to study complex functions and their properties. By focusing on angles rather than distances, conformal geometry has applications in various areas, including theoretical physics, computer graphics, and cartography.