Recent developments in precision mass spectrometry of radioactive isotopes (RI) and some selected related physics subjects are reviewed. In the last decades, besides conventional technologies in RI beam experiments, mass spectrometry of short-lived nuclei has significantly boosted its performance in terms of sensitivity and precision. Whereas single-path measurements are still employed for studies of the most rarely produced short-lived nuclides, though at a moderate precision level, modern methods are based on the storage of freshly produced exotic nuclei for a period of time. One of the biggest achievements in this direction so far is the Penning-trap mass spectrometry, where the highest mass accuracies of typically $10^{-8}$ are routinely obtained for radioactive nuclei. New precise mass values contribute to the developments of nuclear structure and nuclear astrophysics as well as are used for testing fundamental interactions and symmetries. Having the goal to study the most exotic short-lived nuclei in the vicinity of the neutron and proton drip lines, new highly efficient and fast techniques, such as multi-reflection time-of-flight spectrometers (MR-TOF) and mass spectrometry based on heavy-ion storage rings, are steadily gaining importance. Thanks to their fast measurement schemes both, MR-TOFs and storage rings, have successfully proven their high potential by accessing short-lived nuclei, where the mass accuracies down to $10^{-7}$ and even below were reported. Currently, the MR-TOF systems dedicated to mass measurements are in operation or in a planning phase at basically all RI beam facilities. The storage rings for radioactive beams are in use at three accelerator complexes, namely GSI in Darmstadt, Germany, IMP in Lanzhou, P. R. China, and RIKEN in Saitama, Japan. Although the progress of RI beam facilities was enormous in the last decades and new, more powerful facilities are expected to come in operation in the coming few years, some regions on the nuclidic chart will remain inaccessible for experiments. Therefore, the properties of such nuclides, in particular heavy neutron-rich nuclei, will have to be determined through theoretical calculations. It is an important quest to use the new precision masses for constraining and further developing of reliable nuclear theory. There are various models to describe atomic (nuclear) ground-state masses ranging from macroscopic approaches with microscopic corrections to ab-initio calculations based on nucleon–nucleon interactions. In this review article we focus on recent experimental challenges and findings as well as on the cutting-edge experimental technologies and mass formula theories.