A complex manifold is a mathematical structure that generalizes the concept of surfaces to higher dimensions, allowing for complex numbers to be used in defining its properties. It consists of a set of points that locally resemble complex Euclidean space, meaning that around each point, there is a neighborhood that can be described using complex coordinates. These manifolds are essential in various fields, including algebraic geometry and string theory, as they provide a framework for studying complex shapes and their properties. The study of complex manifolds involves understanding their topology, geometry, and the behavior of functions defined on them.