Low doses of ionizing radiation are widely believed to produce effects similar to those observed at high doses; only the incidence (i.e. risk) varies with dose. Furthermore, it is assumed that effects other than those observed at high doses will not occur at low doses. Yet there have been frequent reports in the literature of "anomalies" at low doses--effects unrelated to and unpredictable from the high-dose exposure experiences. These have been referred to as "hormetic" effects. These effects have their parallel with other hormetic effects seen with many other agents generally considered toxic. It is postulated that hormesis has previously received scant attention because it conflicts with the conventional radiation science paradigm.