The theory of characteristic classes is a mathematical framework used in differential geometry and topology to study vector bundles. Characteristic classes provide a way to associate algebraic invariants to these bundles, which can help classify them and understand their geometric properties. These classes are often represented by cohomology classes, which are tools in algebraic topology that capture information about the shape and structure of spaces. Important examples of characteristic classes include the Chern classes, Stiefel-Whitney classes, and Pontryagin classes, each serving different purposes in the study of manifolds and vector bundles.