Theoretical understanding of the nuclear incompressibility: where do we stand ?

The status of the theoretical research on the compressional modes of finite nuclei and the incompressibility $K_{\mathrm{\infty }}$ of nuclear matter, is reviewed. It is argued that the recent experimental data on the Isoscalar Giant Monopole Resonance (ISGMR) allow extracting the value of $K_{\mathrm{\infty }}$ with an uncertainity of about $\pm$ 12 MeV. Non-relativistic (Skyrme, Gogny) and relativistic mean field models predict for $K_{\mathrm{\infty }}$ values which are significantly different from one another, namely $\approx$ 220-235 and $\approx$ 250-270 MeV respectively. It is shown that the solution of this puzzle requires a better determination of the symmetry energy at, and around, saturation. The role played by the experimental data of the Isoscalar Giant Dipole Resonance (ISGDR) is also discussed.