  Supersymmetric models provide very interesting scenarios to account for the
dark matter of the Universe. In this talk we discuss scenarios with gravitino
dark matter in R-parity breaking vacua, which not only reproduce very naturally
the observed dark matter relic density, but also lead to a thermal history of
the Universe consistent with the observed abundances of primordial elements and
the observed matter-antimatter asymmetry. In this class of scenarios the dark
matter gravitinos are no longer stable, but decay with very long lifetimes into
Standard Model particles, thus opening the possibility of their indirect
detection. We have computed the expected contribution from gravitino decay to
the primary cosmic rays and we have found that a gravitino with a mass of 150
GeV and a lifetime of 10^26 s could simultaneously explain the EGRET anomaly in
the extragalactic gamma-ray background and the HEAT excess in the positron
fraction.
