Let V be a real algebraic variety and X be the space of its complex points provided with the involution induced by complex conjugation. Some 50 years ago, Atiyah introduced Real K-theory as a tool to investigate algebraic invariants of V. The purpose of this work is to pursue this study in the framework of algebraic Witt groups. We prove for instance that the Witt group W(V) is "almost" isomorphic to a Real topological analog constructed via Bott periodicity. This is joint work with M. Schlichting and C. Weibel.