  We present XMM-Newton and Chandra X-ray observations of the middle-aged radio
pulsar PSR B0355+54. Our X-ray observations reveal emission not only from the
pulsar itself, but also from a compact diffuse component extending ~50'' in the
opposite direction to the pulsar's proper motion. There is also evidence for
the presence of fainter diffuse emission extending ~5' from the point source.
The compact diffuse feature is well-fitted with a power-law, the index of which
is consistent with the values found for other pulsar wind nebulae. The
morphology of the diffuse component is similar to the ram-pressure confined
pulsar wind nebulae detected for other sources. The X-ray emission from the
pulsar itself is described well by a thermal plus power-law fit, with the
thermal emission most likely originating in a hot polar cap.
