The Pioneer 10/11 spacecraft yielded the most precise navigation in deepspace to date. However, their radio-metric tracking data has consistentlyindicated the presence of a small, anomalous, Doppler frequency drift. Thedrift is a blue-shift, uniformly changing with a rate of ~6 × 10-9 Hz/s and can be interpreted as a constant sunward acceleration ofeach particular spacecraft of aP = (8.74 ± 1.33) × 10-10 m/s2. The nature of this anomaly remains unexplained. Here wesummarize our current knowledge of the discovered effect and review some ofthe mechanisms proposed for its explanation. Currently we are preparing forthe analysis of the entire set of the available Pioneer 10/11 Doppler datawhich may shed a new light on the origin of the anomaly. We present apreliminary assessment of such an intriguing possibility.