Optical waveguides have assumed major importance, not only in optoelectronic applications but also, quite recently, in a study of fundamental physical properties of materials. The propagation characteristics of linear optical waveguides, and to a lesser extent those of waveguides curved in a single plane, are well understood. However, optical waveguides having three-dimensional curvature, for example the helical waveguide, have been proposed and fabricated and an analysis of its properties is essential. In this paper the scalar wave equations for a three-dimensionally curved optical fibre are solved analytically and boundary conditions are applied to the curved core-cladding interface. An asymptotic formula for computing the propagation constant is proposed and the effects of curvilinearity on the characterization of the bound modes are discussed. By using the equivalent fictitious electromagnetic current method, the far radiative field is established and expressions for the radiation losses for various modes are derived.