Since 1985, a focused earthquake prediction experiment has been in progress along the San Andreas fault near the town of Parkfield in central California. Parkfield has experienced six moderate earthquakes since 1857 at average intervals of 22 years, the most recent a magnitude 6 event in 1966. The probability of another moderate earthquake soon appears high, but studies assigning it a 95% chance of occurring before 1993 now appear to have been oversimplified. The identification of a Parkfield fault "segment" was initially based on geometric features in the surface trace of the San Andreas fault, but more recent microearthquake studies have demonstrated that those features do not extend to seismogenic depths. On the other hand, geodetic measurements are consistent with the existence of a "locked" patch on the fault beneath Parkfield that has presently accumulated a slip deficit equal to the slip in the 1966 earthquake. A magnitude 4.7 earthquake in October 1992 brought the Parkfield experiment to its highest level of alert, with a 72-hour public warning that there was a 37% chance of a magnitude 6 event. However, this warning proved to be a false alarm. Most data collected at Parkfield indicate that strain is accumulating at a constant rate on this part of the San Andreas fault, but some interesting departures from this behavior have been recorded. Here we outline the scientific arguments bearing on when the next Parkfield earthquake is likely to occur and summarize geophysical observations to date.