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The classical theory of Sturm sequences provides an algorithm to determine the number of roots of a polynomial with real coefficients contained in a given interval. The main object of this thesis is to show that an adequate generalization of the theory of Sturm sequences provides among other things: a notion of Maslov index for an algebraic loop of Lagrangians defined on a commutative ring; a proof of the fundamental theorem of Hermitian (algebraic) K-theory, theorem due to M. Karoubi; a proof of Bott's periodicity theorems (topological), in the spirit of the work of F. Latour; a symplectic-linear computation of the relative group K2 for all commutative rings, in the spirit of the work of R. Sharpe. The book is as far as possible "self-contained" and elementary: it essentially implements linear or bilinear algebra arguments. It presents a unified approach to the Maslov index in terms of Sturm sequences and quadratic forms.