Like few other parameters, the mass of an atom and its inherent connection with the atomic and nuclear binding energy is a fundamental property, a unique fingerprint of the atomic nucleus. Each nuclide comes with its own mass value different from all others. For short-lived exotic atomic nuclei the importance of its mass ranges from the verification of nuclear models to a test of the Standard Model, in particular with regard to the weak interaction and the unitarity of the Cabibbo–Kobayashi–Maskawa quark mixing matrix. In addition, accurate mass values are important for a variety of applications that extend beyond nuclear physics. Mass measurements on stable atoms now reach a relative uncertainty of 10-11. This extreme accuracy contributes, among other things, to metrology, for example the determination of fundamental constants and a new definition of the kilogram, and to tests of quantum electrodynamics, E=mc2, and fundamental charge, parity, and time reversal symmetry [1]. The introduction of Penning traps into the field of mass spectrometry has made this method a prime choice for high-accuracy measurements on short-lived and stable nuclides. This is reflected in the large number of traps in operation, under construction, or planned world-wide. With the development and application of proper cooling and detection methods the trapping technique has the potential to provide the highest sensitivity and accuracy, even for very short-lived nuclides far from stability. This contribution describes the basics and recent progress made in ion trapping, cooling, and detection for high-accuracy mass measurements with Penning traps. Special attention is devoted to the applications of accurate mass values in different fields of physics.