A 3-manifold is a mathematical space that locally resembles three-dimensional Euclidean space. This means that if you zoom in on any small part of a 3-manifold, it looks like ordinary three-dimensional space, even if the overall shape is more complex. Examples of 3-manifolds include the surface of a sphere and a torus, which can be visualized as a doughnut shape. In topology, the study of shapes and spaces, 3-manifolds are important for understanding the properties of different geometric structures. They can be classified based on their characteristics, such as whether they are compact or non-compact, and whether they have boundaries. The famous Poincaré Conjecture, proven by mathematician Grigori Perelman, is a significant result in the study of 3-manifolds.