In modelling any physical situation, a balance must be struck between making enough assumptions to give a mathematically trac table problem, yet sufficiently few assumptions for the model to remain physically realistic. In this thesis we consider three models which have been proposed for radiation losses in bent fibre optic waveguides and put forward a model of our own which will accommodate a fibre with a W-shaped refractive index profile of a type currently encountered in industrial production. In Chapter 1 we review the original model of Kath and Kriegsmann. We describe in Chapter 2 an idealised ordinary differential equation model due to Paris and Wood and its ad aptation by Burzlaff and Wood to step-function profiles Neither of these idealised models will handle the the realistic W-shaped profile. Chapter 3 contains new work whereby, following the approach of Burzlaff and Wood, we construct a model which incorporates m the boundary condition various geometrical parameters describing the W-shaped profile. The exponentially small imaginary p a rt of the eigenvalue of the resulting boundary value problem corresponds physically to the rate of radiation loss from the fibre. To solve this problem we use a new general method of Hu and Cheng, which relies on concepts introduced by Gmgold These are outlined in Chapter 4 Chapter 5 starts with a rederivation of the formula of Hu and Cheng for the imaginary part of the eigenvalue for general potentials. For our model of Chapter 3 the method of Hu and Cheng can be simplified and we obtain an asymptotic estimate of the eigenvalue based on Hankel function solutions of the differential equation. This is the mam result of the thesis. We conclude by showing th a t the general formula of Hu and Cheng yields th e correct approximation for the rate of radiation loss in the power index models considered by Brazel, Lawless, Liu and Wood.