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The triplet or type-II seesaw mechanism is the simplest way to endow neutrinos with mass in the Standard Model (SM). Here we review its associated theory and phenomenology, including restrictions from $S$, $T$, $U$ parameters, neutrino experiments, charged lepton flavour violations as well as collider searches. We also examine restrictions coming from requiring consistency of electroweak symmetry breaking, i.e. perturbative unitarity and stability of the vacuum. Finally, we discuss novel effects associated to the scalar mediator of neutrino mass generation namely, (i) rare processes, e.g. $l_α\to l_βγ$ decays, at the intensity frontier, and also (ii) four-lepton signatures in colliders at the high-energy frontier. These can be used to probe neutrino properties in an important way, providing a test of the absolute neutrino mass and mass-ordering, as well as of the atmospheric octant. They may also provide the first evidence for charged lepton flavour violation in nature. In contrast, neutrino non-standard interaction strengths are found to lie below current detectability.