Results are presented of an extensive X-ray survey of 380 giant and supergiant stars of spectral types from F to M, carried out with the Einstein Observatory. It was found that the observed F giants or subgiants (slightly evolved stars with a mass M less than about 2 solar masses) are X-ray emitters at the same level of main-sequence stars of similar spectral type. The G giants show a range of emissions more than 3 orders of magnitude wide; some single G giants exist with X-ray luminosities comparable to RS CVn systems, while some nearby large G giants have upper limits on the X-ray emission below typical solar values. The K giants have an observed X-ray emission level significantly lower than F and F giants. None of the 29 M giants were detected, except for one spectroscopic binary. 79 refs.