Pin overview of some recent new achievements in nuclear moment measurements on nuclei produced and spin-oriented in fragmentation reactions is presented. By combining different types of spin-orientation with different experimental techniques, we have shown that the magnetic dipole moments and electric quadrupole moments of exotic nuclei in their ground state as well as in their excited isomeric states can be measured. The g-factor of the neutron rich Al-31 ground state has been measured by using for the first time a spin-polarized beam at the LISE3 spectrometer at GANIL. Isomer g-factors in the Ni-68 region have been measured by detecting the Larmor precession of a spin-aligned ensemble of selected isomers. Preliminary results are discussed.