It is well established that an earthquake in the Earth's crust can trigger subsequent earthquakes, but such triggering has not been documented for deeper earthquakes. Models for shallow fault interactions suggest that static (permanent) stress changes can trigger nearby earthquakes, within a few fault lengths from the causative earthquake, whereas dynamic (transient) stresses carried by seismic waves may trigger earthquakes both nearby and at remote distances. Here we present a detailed analysis of the 19 August 2002 Tonga deep earthquake sequences and show evidence for both static and dynamic triggering. Seven minutes after a magnitude 7.6 earthquake occurred at a depth of 598 km, a magnitude 7.7 earthquake (664 km depth) occurred 300 km away, in a previously aseismic region. We found that nearby aftershocks of the first mainshock are preferentially located in regions where static stresses are predicted to have been enhanced by the mainshock. But the second mainshock and other triggered events are located at larger distances where static stress increases should be negligible, thus suggesting dynamic triggering. The origin times of the triggered events do not correspond to arrival times of the main seismic waves from the mainshocks and the dynamically triggered earthquakes frequently occur in aseismic regions below or adjacent to the seismic zone. We propose that these events are triggered by transient effects in regions near criticality, but where earthquakes have difficulty nucleating without external influences.