Algebraic geometry is the study of varieties - objects which locally look like the zero sets of polynomials. This definition is analogous to that of a smooth manifold - an object which locally looks like Euclidean space. However, the global structure of varieties and that of smooth manifolds diverge in important ways. I will show how vector bundles help elucidate this global structure and how these ideas connect to fundamental questions in representation theory, number theory and algebra. To conclude, we sketch how related ideas can be used to settle an old question of Grothendieck on etale cohomology.