Named after the Pioneer 10 and 11 space probes, the Pioneer anomaly refers to the fact that they seem to be moving a teensy bit different from how we think they should be moving (or, more technically, the spacecraft seem to be subject to an unmodeled acceleration whose direction is towards the Sun).
The anomaly was first noticed, by John Anderson, in 1980, when analysis of tracking data from the spacecraft showed a small, unexplained acceleration towards the Sun (this was first published in 1995, with the main paper appearing in 1998). Since then it has been studied continuously, by quite a few scientists.
The Pioneer anomaly is one of the (very few!) true mysteries in contemporary physics, and is a great example of how science is done.
The first step – which Anderson and colleagues took – was to work out where the spacecraft were, and how fast they were traveling (and in what direction), at as many times as they could. Then they estimated the effects of gravity, from all known solar system objects (from the Sun to tiny asteroids and comets). Then they estimated the effects of things like radiation pressure, and possible outgassing. Then … They also checked whether other spacecraft seemed to have experienced a similar anomalous acceleration (the net: not possible to get an unambiguous answer, because all others have known – but unmodelable – effects much bigger than the Pioneer anomaly). Several independent investigations have been conducted, using different approaches, etc.
In the last few years, much effort has gone into trying to find all the raw tracking data (this has been tough, many tapes have been misplaced, for example), and into extracting clean signals from this (also tough … the data were never intended to be analyzed this way, meta-data is sorely lacking, and so on).
And yet, the anomaly remains …
… there’s an unmodeled acceleration of approximately 9 x 10-10 m/s2, towards the Sun.
The Planetary Society has been funding research into the Pioneer anomaly, and has a great summary here! And you can be a fly on the wall at a meeting of a team of scientists investigating the Pioneer anomaly, by checking out this Pioneer Explorer Collaboration webpage.
Universe Today has several stories on the Pioneer anomaly, for example The Pioneer Anomaly: A Deviation from Einstein Gravity?, Is the Kuiper Belt Slowing the Pioneer Spacecraft?, and Ten Mysteries of the Solar System.
The Planetary Society