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Penning-trap mass spectrometry in atomic and nuclear physics has become a well-established and reliable tool for the determination of atomic masses. In combination with short-lived radioactive nuclides it was first introduced at ISOLTRAP at the Isotope Mass Separator On-Line facility (ISOLDE) at CERN. Penning traps have found new applications in coupling to other production mechanisms, such as in-flight production and separation systems. The applications in atomic and nuclear physics range from nuclear structure studies and related precision tests of theoretical approaches to description of the strong interaction to tests of the electroweak Standard Model, quantum electrodynamics and neutrino physics, and applications in nuclear astrophysics. The success of Penning-trap mass spectrometry is due to its precision and accuracy, even for low ion intensities (i.e., low production yields), as well as its very fast measurement cycle, enabling access to short-lived isotopes. The current reach in relative mass precision goes beyond δ m/ m=10 −8 , the half-life limit is as low as a few milliseconds, and the sensitivity is on the order of one ion per minute in the trap. We provide a comprehensive overview of the techniques and applications of Penning-trap mass spectrometry in nuclear and atomic physics.