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Percolation and synchronization are two phase transitions that have beenextensively studied since already long ago. A classic result is that, in thevast majority of cases, these transitions are of the second-order type, i.e.continuous and reversible. Recently, however, explosive phenomena have beenreported in com- plex networks' structure and dynamics, which rather remindfirst-order (discontinuous and irreversible) transitions. Explosivepercolation, which was discovered in 2009, corresponds to an abrupt change inthe 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 phaseoscillators were taken into consideration. The two phenomena have stimulatedinvestigations and de- bates, attracting attention in many relevant fields. Sofar, various substantial contributions and progresses (including experimentalverifications) have been made, which have provided insights on what structuraland dynamical properties are needed for inducing such abrupt transformations,as well as have greatly enhanced our understanding of phase transitions innetworked systems. Our intention is to offer here a monographic review on themain-stream literature, with the twofold aim of summarizing the existingresults and pointing out possible directions for future research.

Explosive transitions in complex networks' structure and dynamics: percolation and synchronization