Over the years, non-spreading or non-diffracting wave configurations have been systematically investigated in optics. Perhaps the best known example of a diffraction-free optical wave is the so-called Bessel beam, first suggested and observed by Durnin et al. This work sparked considerable theoretical and experimental activity and paved the way toward the discovery of other interesting non-diffracting solutions. In 1979 Berry and Balazs made an important observation within the context of quantum mechanics: they theoretically demonstrated that the Schrodinger equation describing a free particle can exhibit a non-spreading Airy wavepacket solution. This work remained largely unnoticed in the literature-partly because such wavepackets cannot be readily synthesized in quantum mechanics. In this dissertation we investigate both theoretically and experimentally the acceleration dynamics of non-spreading optical Airy beams in both one- and two-dimensional configurations. We show that this class of finite energy waves can retain their intensity features over several diffraction lengths. The possibility of other physical realizations involving spatio-temporal Airy wavepackets is also considered. As demonstrated in our experiments, these Airy beams can exhibit unusual features such as the ability to remain quasi-diffraction-free over long distances while their intensity features tend to freely accelerate during propagation. We have demonstrated experimentally that optical Airy beams propagating in free space can perform ballistic dynamics akin to those of projectiles moving under the action of gravity. The parabolic trajectories of these beams as well as the motion of their center of gravity were observed in good agreement with theory. Another remarkable property of optical Airy beams is their resilience in amplitude and phase perturbations. We show that this class of waves tends to reform during propagation in spite of the severity of the imposed perturbations. In all occasions the reconstruction of these beams is interpreted through their internal transverse power flow. The robustness of these optical beams in scattering and turbulent environments was also studied. The experimental observation of self-trapped Airy beams in unbiased nonlinear photorefractive media is also reported. This new class of non-local self-localized beams owes its existence to carrier diffusion effects as opposed to self-focusing. These finite energy Airy states exhibit a highly asymmetric intensity profile that is determined by the inherent properties of the nonlinear crystal. In addition, these wavepackets self-bend during propagation at an acceleration rate that is independent of the thermal energy associated with two-wave mixing diffusion photorefractive nonlinearity.