In this course, we study real and complex vector bundles, their characteristic classes and various applications. These include Stiefel-Whitney classes, Chern classes, Pontrjagin classes, the splitting principle, classifying spaces, Grassmanians, Hirzebruch's signature theorem, and the 7-dimensional exotic spheres of Milnor. Additional topics may include Morse Theory, the h-cobordism Theorem, and the Poincare Conjecture in higher dimensions.