Magnetic moments have in general provided substantial information on the microscopic structure of nuclear states. Advances in technology have made it possible to measure, with reasonable accuracy, magnetic moments of nuclear states with lifetimes of the order of picoseconds or less. Improvements over the last decade involve all experimental aspects ranging from the availability of a wide variety of stable and radioactive beams, to the realization of highly segmented, high resolution detector arrays. A microscopic understanding of the hyperfine interaction between the nuclear moments and the ferromagnetic environment with which they interact has not yet been reached. Nevertheless, elegant applications of the new experimental techniques have allowed the measurement of nuclear moments of many nuclear states that were not accessible previously. Such measurements have shown how these moments vary with energy, spin and temperature in specific nuclei, or across isotopic chains. New results, in turn, have placed stringent constraints on theoretical calculations. The technical advances and selected highlights of relevant measurements across the periodic table are described in this review. Recent examples of applications of transient field and recoil-in-vacuum techniques in a radioactive beam environment are presented.