  ROSAT observations found no convincing evidence for X-ray sources located in
local dwarf spheroidal galaxies (dSph). Now with more sensitive instruments on
board Chandra and XMM-Newton we can reach fainter luminosity levels. We report
on an observation of the Sagittarius (Sgr) dSph made using Chandra and an
observation of the Carina (Car) dSph made using XMM-Newton. Our observations
are sensitive to sources with X-ray luminosities in the 0.1-10 keV band of
\~1x10^{32 ergs/s and 3x10^34 ergs/s for the Sgr and Car fields respectively.
We have identified a total of 80 sources in the Sgr field and 53 sources in the
Car field. Although the source numbers are roughly consistent with the expected
number of background AGN, we found a small fraction of X-ray sources which were
soft and could be located in the host dSph. Follow-up optical/IR observations
may help to identify their optical counterparts and hence determine their
nature.
