The classical theory of Sturm sequences provides an algorithm for determining the number of roots of a polynomial with real coefficients contained in an open interval. The main purpose of this monograph is to show that a suitable generalization of the theory of Sturm sequences provides, among others: a notion of Maslov index for an algebraic loop of lagrangians defined over a commutative ring; a proof of the fundamental theorem of (algebraic) hermitian K-theory (theorem due to M. Karoubi); a proof of the theorems of (topological) Bott periodicity (in the spirit of the work of F. Latour); the...