Scalarization is a mechanism that endows strongly self-gravitating bodies,
such as neutron stars and black holes, with a scalar field configuration. It
resembles a phase transition in that the scalar configuration only appears when
a certain quantity that characterizes the compact object, e.g., its compactness
or spin, is beyond a threshold. We provide a critical and comprehensive review
of scalarization, including the mechanism itself, theories that exhibit it, its
manifestation in neutron stars, black holes, and their binaries, potential
extension to other fields, and a thorough discussion of future perspectives.