A smooth manifold is a mathematical space that resembles Euclidean space but can have a more complex structure. It allows for the concept of smoothness, meaning that you can perform calculus on it. Smooth manifolds are defined by charts, which are mappings from open sets of the manifold to open sets of Euclidean space, ensuring that transitions between charts are smooth. These manifolds are essential in various fields, including physics, where they help describe the shape of spacetime, and in differential geometry, where they provide a framework for studying curves and surfaces. They serve as a foundation for understanding more complex structures in mathematics and science.