Airy beams  are beams whose transverse amplitude dependence at origin is defined by the Airy function. These beams have several unique properties: They are termed "non-diffracting" since the infinite Airy wavepacket does not spread as it propagates, and accelerating, since this wavepacket is centered around a parabolic trajectory in space. These unique features are still observed to a large extent with truncated Airy beams, in which the Airy function distribution is multiplied by either a Gaussian or an exponential window. Possible applications of Airy beams include optical micro-manipulation of small particles, generation of curved plasma channels in air and generation of Airy-Bessel light bullets. Moreover, the Airy beam is only one example of an accelerating beam: Accelerating parabolic beams are non-diffracting and freely accelerating beams that form an orthogonal and complete set of solutions to the paraxial wave equation.