A review of Josephson shift-register circuits that have been designed, fabricated, or tested is presented with emphasis on work in the 1980s. Operating speed is most important, since it often limits system performance. Older designs used square-wave clocks, but most modern designs use offset sine waves, with either two or three phases. Operating margins and gate bias uniformity are key concerns. The fastest measured Josephson shift register operated at 2.3 GHz, which compares well with a GaAs shift register that consumes 250 times more power. The difficulties of high-speed testing have prevented many Josephson shift registers from being operated at their highest speeds. Computer simulations suggest that 30-GHz operation is possible with current Nb/Al/sub 2/O/sub 3//Nb technology. Junctions with critical current densities near 10 kA/cm/sup 2/ would make 100-GHz shift registers feasible.>