  We develop the concept of ``dragon-kings'' corresponding to meaningful
outliers, which are found to coexist with power laws in the distributions of
event sizes under a broad range of conditions in a large variety of systems.
These dragon-kings reveal the existence of mechanisms of self-organization that
are not apparent otherwise from the distribution of their smaller siblings. We
present a generic phase diagram to explain the generation of dragon-kings and
document their presence in six different examples (distribution of city sizes,
distribution of acoustic emissions associated with material failure,
distribution of velocity increments in hydrodynamic turbulence, distribution of
financial drawdowns, distribution of the energies of epileptic seizures in
humans and in model animals, distribution of the earthquake energies). We
emphasize the importance of understanding dragon-kings as being often
associated with a neighborhood of what can be called equivalently a phase
transition, a bifurcation, a catastrophe (in the sense of Rene Thom), or a
tipping point. The presence of a phase transition is crucial to learn how to
diagnose in advance the symptoms associated with a coming dragon-king. Several
examples of predictions using the derived log-periodic power law method are
discussed, including material failure predictions and the forecasts of the end
of financial bubbles.
