In this review we give an overview of the present status of research on carbon nanotube (CNT) field emitters and their applications. Several different construction principles of field–emission devices with CNTs are summarized. The emission mechanism is introduced and a detailed overview is given of the measured emission properties and related topics of CNT electron sources. We give also several examples of field–emission devices with CNT electron emitters that are presently being investigated in the academic world as well as in industry. Carbon nanotube electron sources clearly have interesting properties, such as low voltage operation, good stability, long lifetime and high brightness. The most promising applications are the field–emission display and high–resolution electron–beam instruments. But several hurdles remain, such as the manufacture of an electron source or an array of electron sources with exactly the desired properties in a reproducible manner.