The theory of Stokes flows about slender particles is considered for the case in which the particle cross-section is arbitrary and non-uniform along the axis of the particle, subject to certain smoothness assumptions. As distinct from previous works on slender particles in which the particle is represented approximately by a line distribution of Stokeslets, the representation adopted here is the exact one of a surface distribution of Stokeslets. This representation may then be used to recover the familiar one-dimensional integral equation for the equivalent line density of Stokeslets, together with estimates for the range of particle shapes for which the equation is valid. Within this range it is found that there is a class of particles for which the established perturbation scheme, which is used to obtain a solution to the integral equation, is singular. This class of particle shapes is illustrated by the example of a uniformly twisted particle whose pitch of twist is large enough compared with the cross-sectional dimension to ensure the validity of the equation, but small enough to make the usual method of solution singular. It is shown how the equation may be transformed so that an approximate solution can be found by means of a regular perturbation scheme. The results indicate that, for each of the axial and transverse components of motion, there is an equivalent particle of circular cross-section for which the total force and couple are the same as for the original particle. However, the radii of the equivalent cylinders are different for each component, the transverse component being affected by the twist while the axial component is not.