The induction accelerator was conceived by N. C. Christofilos and first realized as the Astron accelerator that operated at LLNL from the early 1960's to the end of 1975. This accelerator generated electron beams at energies near 6 MeV with typical currents of 600 Amperes in 400 ns pulses. The Advanced Test Accelerator (ATA) built at Livermore's Site 300 produced 10,000 Ampere beams with pulse widths of 70 ns at energies approaching 50 MeV. Several other electron and ion induction accelerators have been fabricated at LLNL and LBNL. This paper reviews the principal diagnostics developed through efforts by scientists at both laboratories for measuring the current, position, energy, and emittance of beams generated by these high current, short pulse accelerators. Many of these diagnostics are closely related to those developed for other accelerators. However, the very fast and intense current pulses often require special diagnostic techniques and considerations. The physics and design of the more unique diagnostics developed for electron induction accelerators are presented and discussed in detail.