In the work, we describe the theory of characteristic classes of vector bundles and apply it to formulate and prove Bott vanishing theorem. We provide a detailed presentation of the theory of real and complex vector bundles, which lays the foundation for the entire work. Special emphasis is placed on constructions, metrics, and short exact sequences of vector bundles. We define connection and curvature on bundles, describe their properties, and present constructions of connections on vector bundles. Next tool that we present in this work is de Rham cohomology and its homotopy invariance. As Bott's theorem deals with foliations, we briefly examine the theory of foliations with a special focus on studying the subbundles of the tangent bundle using differential forms. Before defining characteristic classes, we also explore the theory of invariant polynomials and their connection to symmetric polynomials. Then, using the developed tools and the Chern-Weil homomorphism, we define real and complex characteristic classes. We take a special look at the Pontryagin and Chern classes and their properties, and finally we present Bott's theorem and its proof.
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