The theoretical prediction that stars that develop heavy oxygen cores will become
pair-unstable and explode has been made many decades ago. Yet, for many years, no
examples of such explosions were found, and it was often conjectured that stars
massive enough to explode in this manner may only exist at very high redshifts
(population III stars). In recent years several luminous supernova explosions have
been detected, and some of them have been proposed to result from this mechanism. I
will review the theoretical predictions for pair-instability supernovae and confront
them with observations, showing that indeed at least one object, SN 2007bi, is a
convincing example of a pair-instability supernova. I will describe additional
examples recently detected both nearby and in the distant Universe, and comment about
the implications for stellar evolution models, the chemical evolution of the
Universe, and future detection of the first stars.