A field emission electron gun has a field emission cathode composed of a carbon fiber and a conductive base for supporting the carbon fiber, an extractor for causing the field emission of electrons, and an accelerator for accelerating the electrons. The carbon fiber contains at least one of trivalent and pentavalent elements. In particular, the trivalent and pentavalent elements are boron and/or nitrogen. The content of at least one of boron and nitrogen in the carbon fiber is 0.1% to 5% at an atomic weight ratio of the contained element to carbon. The diameter of the carbon fiber is 20 nm to 200 nm. Such an electron gun can have a high-brightness electron beam with a narrow energy width. The field emission electron gun is applied to various electron beam apparatus.