Previous observations of the middle-aged pulsar Geminga with XMM-Newton and Chandra have shown an unusual pulsar wind nebula (PWN), with a 20 long central (axial) tail directed opposite to the pulsars proper motion and two 2 long, bent lateral (outer) tails. Here we report on a deeper (78 ks) Chandra observation and a few additional XMM-Newton observations of the Geminga PWN. The new Chandra observation has shown that the axial tail, which includes up to three brighter blobs, extends at least 50 (i.e., 0.06 d_{250} pc) from the pulsar. It also allowed us to image the patchy outer tails and the emission in the immediate vicinity of the pulsar with high resolution. The PWN luminosity, L_{0.3-8 keV} ~ 3times 10^{29} d_{250}^2 erg/s, is lower than the pulsars magnetospheric luminosity by a factor of 10. The spectra of the PWN elements are rather hard (photon index ~ 1). Comparing the two Chandra images, we found evidence of PWN variability, including possible motion of the blobs along the axial tail. The X-ray PWN is the synchrotron radiation from relativistic particles of the pulsar wind; its morphology is connected with the supersonic motion of Geminga. We speculate that the outer tails are either (1) a sky projection of the limb-brightened boundary of a shell formed in the region of contact discontinuity, where the wind bulk flow is decelerated by shear instability, or (2) polar outflows from the pulsar bent by the ram pressure from the ISM. In the former case, the axial tail may be a jet emanating along the pulsars spin axis, perhaps aligned with the direction of motion. In the latter case, the axial tail may be the shocked pulsar wind collimated by the ram pressure.
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