Since Bott proved his famous periodicity theorems in 1959, a lot of work has been done in order to ``understand" them from various points of view. In this lecture, we make a tour of about 45 years of mathematics, starting from the topological/analytic viewpoint (Grothendieck, Atiyah, Hirzebruch) and ending with recent achievements in algebraic K-theory. A written version of this lecture may be found in the following Web site : http://www.math.uiuc.edu/~dan/KtheoryHandbook/Karoubi/Bott_per.pdf