Apparent velocities greater than the velocity of light are of course optical illusions and contain no physics which would not be compatible with the theory of special relativity. The explanation can be given in a fairly straightforward way as a light travel time effect. Actual derived velocities, however, are close to the speed of light and give an interesting example of special relativistic effects (such as Doppler shifts) observed by a bulk of mass moving close to the velocity of light. They are often seen in two opposing jets, one moving away and one moving toward us. If in both sources Doppler shifts are observed, the velocity 'and' the distance can be determined independent of other observations.
In 1966 Sir Martin Rees predicted (Nature 211, 468) that "an object moving relativistically in suitable directions may appear to a distant observer to have a transverse velocity much greater than the velocity of light".
A few years later (in 1970) such sources were indeed discovered as very distant astronomical radio sources, such as radio galaxies and quasars. They were called superluminal sources (litt. "faster than light") sources. The discovery was a spectacular result of a new technique called Very Long Baseline Interferometry, which allowed to determine positions better than milli-arcseconds and in particular to determine the change in positions on the sky, called proper motions in a timespan of typically years. The apparent velocity is obtained by multiplying the observed proper motion by the distance and could be up to 6 times the speed of light.
In 1994 a galactic speed record was obtained with the discovery of a superluminal source in our own galaxy, the cosmic x-ray source GRS1915+105. The expansion occurred on a much shorter timescale. Several separate blobs were seen (I.F. Mirabel and L.F. Rodriguez, Nature 371, 48, "A superluminal source in the Galaxy") to expand in pairs within weeks by typically 0.5 arcsec. Because of the analogy with Quasars, this source was called a microquasar.
''A pair of plasma clouds expelled from the microquasar GRS1915+105. From March 18 1994 to April 16 it moves about 1 arcsec. The source is in the general direction of the galactic center at a distance of 12 kiloparsec =3.7×1020m. At this distance, 1 arcsec corresponds with 1.8×1015 m. The expansion takes place in a month or 2.5×106 s, so one observes an apparent expansion of about 109 m/s, about three times the speed of light. [From the cover of Nature, 1994, vol. 392; Mirabel and Rodriguez, Nature, 392, 673] ''