Recently, Airy beam as a kind of non-diffracting beam, has attracted a great deal of attention due to its unique properties to have propagation-invariant intensity profile, remain transverse accelerating and exhibit self-healing features. Therefore, Airy beams have found many potential applications, such as optical micro-manipulation, imaging technology, surface plasmon polaritons and laser micromachining. Airy optical fiber as a kind of waveguide device can be applied for the Airy beam generation, carry out the exploration of new Airy fiber and expand the Airy beam application range, has important practical significance. In this paper, we give an systematical introduction from the view of the Airy beam working principle, Airy fiber structure design, Airy fiber beam generated internal mechanism, Airy beam wavelength response characteristics, and Airy fiber applications.