  The compression-mode giant resonances, namely the isoscalar giant monopole
and isoscalar giant dipole modes, are examples of collective nuclear motion.
Their main interest stems from the fact that one hopes to extrapolate from
their properties the incompressibility of uniform nuclear matter, which is a
key parameter of the nuclear Equation of State (EoS). Our understanding of
these issues has undergone two major jumps, one in the late 1970s when the
Isoscalar Giant Monopole Resonance (ISGMR) was experimentally identified, and
another around the turn of the millennium since when theory has been able to
start giving reliable error bars to the incompressibility. However, mainly
magic nuclei have been involved in the deduction of the incompressibility from
the vibrations of finite nuclei. The present review deals with the developments
beyond all this. Experimental techniques have been improved, and new
open-shell, and deformed, nuclei have been investigated. The associated changes
in our understanding of the problem of the nuclear incompressibility are
discussed. New theoretical models, decay measurements, and the search for the
evolution of compressional modes in exotic nuclei are also discussed.
