A Riemann surface is a one-dimensional complex manifold, which means it is a shape that can be described using complex numbers. These surfaces allow mathematicians to study multi-valued functions, like the square root or logarithm, in a way that makes them single-valued. By "gluing" together different pieces of the surface, we can create a smooth, continuous structure that reflects the behavior of these functions. Riemann surfaces are named after the mathematician Bernhard Riemann, who contributed significantly to complex analysis. They play a crucial role in various fields, including algebraic geometry and string theory, by providing a framework to understand complex functions and their properties.