Trapped and stored charged particles, atoms and molecules offer a number of opportunities to measure exact values of important fundamental constants such as lepton magnetic anomalies, the fine structure constant and the electron mass. New Physics can be searched for by comparing precise measurements and highly accurate calculations of particle properties. Some recent experiments differ by a few standard deviations from standard theory predictions, such as the muon magnetic anomaly and 21 Na β-decay; for a clarification further work is needed.