Manifold theory is a branch of mathematics that studies spaces called manifolds, which are shapes that can be curved but still resemble flat spaces in small regions. These structures allow mathematicians to generalize concepts from geometry and calculus to more complex forms, making it easier to analyze and understand higher-dimensional spaces. Manifolds can be classified into different types, such as differentiable manifolds, which allow for smooth transitions, and topological manifolds, which focus on the properties that remain unchanged under continuous deformations. This theory has applications in various fields, including physics, particularly in the study of general relativity and string theory.