The source of terrestrial gamma‐ray flashes (TGFs) has remained a mystery since their discovery in 1994. Recent Reuven Ramaty High‐Energy Solar Spectroscopic Imager (RHESSI) observations show that these intense bursts of MeV gamma rays likely originate much deeper in the atmosphere than previously inferred from Burst and Transient Source Experiment (BATSE) data, with the source altitude <21 km. Using existing measurements of the intensity and duration of BATSE and RHESSI TGFs, along with limits on the electric field set by the relativistic feedback mechanism involving backward propagating positrons and x‐rays, it is found that TGFs cannot be produced by relativistic runaway electron avalanches acting on natural background radiation or extensive cosmic‐ray air showers alone, as has been assumed by many previous models. Instead, the energetic seed particle production most likely involves either relativistic feedback or runaway electron production in the strong electric fields associated with lightning leaders or streamers, similar to the energetic radiation observed on the ground from lightning.