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State-of-the-art studies of electromagnetic moments of exotic nuclei in experiments with radioactive beams are presented. These include first results on the 19/2+ isomer in 127Sn, which was populated in a relativistic fragmentation reaction within the g-RISING campaign at GSI, Darmstadt, Germany, and the first measurement of a nuclear quadrupole moment of an isomeric state in fragmentation reaction, that of the 9/2+ isomer in 61Fe, which was performed at GANIL, Caen, France. Results from experiments to measure the electromagnetic moments in transfer reactions are also discussed, as well as the perspectives to utilize this technique in experiments with post-accelerated ISOL beams in inverse kinematics. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]