  We report on two Chandra observations of the 3-Myr pulsar B1929+10, which
reveal a faint compact (~9"x5") nebula elongated in the direction perpendicular
to the pulsar's proper motion, two patchy wings, and a possible short (~3") jet
emerging from the pulsar. In addition, we detect a tail extending up to at
least 4' in the direction opposite to the pulsar's proper motion, aligned with
the 15'-long tail detected in ROSAT and XMM-Newton observations. The overall
morphology of the nebula suggests that the shocked pulsar wind is confined by
the ram pressure due to the pulsar's supersonic speed. The shape of the compact
nebula in the immediate vicinity of the pulsar seems to be consistent with the
current MHD models. However, since these models do not account yet for the
change of the flow velocity at larger distances from the pulsar, they are not
able to constrain the extent of the long pulsar tail. The luminosity of the
whole nebula as seen by Chandra is ~10^30 ergs/s in the 0.3-8 keV band, for the
distance of 361 pc. Using the Chandra and XMM-Newton data, we found that the
pulsar spectrum is comprised of non-thermal (magnetospheric) and thermal
components. The non-thermal component can be described by a power-law model
with photon index ~1.7 and luminosity 1.7x10^30 ergs/s in the 0.3-10 keV band.
The blackbody fit for the thermal component, which presumably emerges from hot
polar caps, gives the temperature kT~0.3 keV and projected emitting area 3x10^3
m^2, corresponding to the bolometric luminosity ~(1-2)x10^30 ergs/s.
