Accelerator mass spectrometry (AMS) was born in the late 1970s, when it was realized at nuclear physics laboratories that the accelerator systems can be used as a sensitive mass spectrometer to measure ultralow traces of long-lived radioisotopes. It soon became possible to measure radioisotope-to-stable-isotope ratios in the range from 10-12 to 10-16 by counting the radioisotope ions "atom by atom"and comparing the count rate with ion currents of stable isotopes (1.6 μA=1×1013 singly charged ions/s). It turned out that electrostatic tandem accelerators are best suited for this, and there are now worldwide about 160 AMS facilities based on this principle. This review presents the history, technological developments, and research areas of AMS through the 45 yr since its discovery. Many different fields are touched by AMS measurements, including archaeology, astrophysics, atmospheric science, biology, climatology, cosmic-ray physics, environmental physics, forensic science, glaciology, geophormology, hydrology, ice core research, meteoritics, nuclear physics, oceanography, and particle physics. Since it is virtually impossible to discuss all fields in detail in this review, only specific fields with recent advances are highlighted in detail. For the others, an effort is made to provide relevant references for in-depth studies of the respective fields.