Extensions centrales dʹalgèbres et de groupes de lie de dimension infinie, algèbre de virasoro et généralisations

This article surveys problems related to central extensions of Lie algebra of vector fields, both from the pure algebraic point of view (cohomological computations) and from the point of view of the geometrical and physical applications. Especially the cases of hamiltonian, contact and unimodular vector fields are developed, including the applications to fluid mechanics.