  We report on the deep (100 ksec) XMM-Newton/EPIC observation of the field of
the Geminga pulsar. The unprecedented throughput of the instrument allowed to
detect two elongated parallel X-ray tails trailing the neutron star. They are
aligned with the object's supersonic motion, extend for ~2', and have a
nonthermal spectrum produced by electron-synchrotron emission in the bow shock
between the pulsar wind and the surrounding medium. Electron lifetime against
synchrotron cooling matches the source transit time over the X-ray features'
length. Such an X-ray detection of a pulsar bow shock allows us to gauge the
pulsar electron injection energy and the shock magnetic field while
constraining the angle of Geminga's motion and the local matter density. We
give also preliminary results on the timing and spectral analysis of the
\~63,000 photons collected from the neutron star.
