Bott periodicity plays an important role in topological K-theory. The purpose of this paper is to extend the periodicity theorem in a discrete context, where all classical groups are involved and not just the general linear group. The present paper generalizes previous results of the author [K1] and [K2], where 2 was assumed to be invertible in the ring. For the proof, two important ideas have to be mentioned. The first one is due to Ranicki [R] who introduced a kind of "enlarged" orthogonal group. The second one is a genuine cup-product between quadratic forms due to Clauwens [C]. As an example of results obtained, we prove that the higher Witt groups of a finite field of characteristic 2 are all isomorphic to Z/2. They generalize in some sense the Dickson and Arf invariants.