By considering the action of the unitary group U(n) on the (2n-1)-dimensional sphere, one can show that the i-th homotopy groups of U(n) stabilize for sufficiently large n. If we denote these (eventually) isomorphic groups by \pi_i(U), the periodicity theorem states that the sequence \pi_i(U) is periodic with period two. In this talk, I will present Bott's original proof (following Milnor's reproduction), which uses only basic elements of homotopy theory and Morse theory.