Pontryagin classes are topological invariants associated with smooth manifolds, particularly in the context of differential geometry and algebraic topology. They arise from the study of vector bundles, specifically the tangent bundle of a manifold. These classes provide important information about the curvature of the manifold and can be used to distinguish between different types of manifolds. The Pontryagin classes are defined using the Chern-Weil theory, which relates differential forms to characteristic classes. They are integral cohomology classes and can be computed from the curvature of a connection on a vector bundle. These classes play a significant role in the study of differential topology and characteristic classes.