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This article provides an insight into several open problems in the quest fornovel modes of excitation in nuclei with isospin asymmetry, deformation andfinite temperature characteristic in stellar environment. Major unsolvedproblems include the nature of pygmy dipole resonances, the quest for variousmultipole and spin-isospin excitations both in neutron-rich and protondrip-line nuclei mainly driven by loosely bound nucleons, excitations inunstable deformed nuclei and evolution of their properties with the shape phasetransition. Exotic modes of excitation in nuclei at finite temperaturescharacteristic for supernova evolution present open problems with possibleimpact in modeling astrophysically relevant weak interaction rates. All theseissues challenge self-consistent many body theory frameworks at the frontiersof on-going research, including nuclear energy density functionals, bothphenomenological and constrained by the strong interaction physics of QCD,models based on low-momentum two-nucleon interaction V_{low-k} and correlatedrealistic nucleon-nucleon interaction V_{UCOM}, supplemented by three-bodyforce, as well as two-nucleon and three-nucleon interactions derived from thechiral effective field theory. Joined theoretical and experimental efforts,including research with radioactive isotope beams, are needed to provideinsight into dynamical properties of nuclei away from the valley of stability,involving the interplay of isospin asymmetry, deformation and finitetemperature.

The quest for novel modes of excitation in exotic nuclei