abstract: Let X be a homogeneous space of a connected linear algebraic group over a field of characteristic zero. Under the assumption that the geometric stabilizers are connected, we show that the Galois module defined by the Picard group of a smooth compactification of X is of a very special type, namely it is a "flasque" Galois lattice. We compute the class of this module up to addition of a permutation module. This recent joint work with B. Kunyavskii extends known results for principal homogeneous spaces under tori (Voskresenskii, Sansuc and the speaker, 1976) and principal homogeneous spaces under connected linear algebraic groups (Borovoi and Kunyavskii, 2004). There are connexions with the Brauer-Manin obstruction and with the study of R-equivalence on rational points.