In this thesis, we focus on three different optical models possessing accelerating solitons, namely, propagation of beams in photorefractive media with diffusion, pulse propagation in fibre links using sliding-frequency filters and pulse propagation governed by the NLS equation with intrapulse Raman scattering. In each case, an ordinary differential equation (ODE) is obtained from the partial differential equation (PDE) through a similarity variable reduction. Then, we investigate parameter ranges for which solutions exist and their normal mode stability. Integration of the PDE is also used to confirm acceleration regime and stability properties. The three accelerating solitons have the common characteristic that the asymptotics of both the ODE and the stability eigenvalue problem involve solutions to Airy equations. This characteristic allows us to use Airy functions and their asymptotic properties in our analytic and numerical evaluations. It also implies that, in using Evans function ideas in a stability analysis, a significant modification is necessary. The asymptotics must be described in terms of appropriate recessive choices of Airy functions. This leads to a novel modification of the Evans function method, which is then applied in the search for eigenvalues, allowing the identification of unstable and internal (oscillatory) modes when they exist. The prediction of these modes is shown to agree with direct numerical integration of the PDEs.