  Percolation and synchronization are two phase transitions that have been
extensively studied since already long ago. A classic result is that, in the
vast majority of cases, these transitions are of the second-order type, i.e.
continuous and reversible. Recently, however, explosive phenomena have been
reported in com- plex networks' structure and dynamics, which rather remind
first-order (discontinuous and irreversible) transitions. Explosive
percolation, which was discovered in 2009, corresponds to an abrupt change in
the network's structure, and explosive synchronization (which is concerned,
instead, with the abrupt emergence of a collective state in the networks'
dynamics) was studied as early as the first models of globally coupled phase
oscillators were taken into consideration. The two phenomena have stimulated
investigations and de- bates, attracting attention in many relevant fields. So
far, various substantial contributions and progresses (including experimental
verifications) have been made, which have provided insights on what structural
and dynamical properties are needed for inducing such abrupt transformations,
as well as have greatly enhanced our understanding of phase transitions in
networked systems. Our intention is to offer here a monographic review on the
main-stream literature, with the twofold aim of summarizing the existing
results and pointing out possible directions for future research.
