Flexion is a non-linear gravitational lensing effect that arises from gradients in the convergence and shear across an image. We derive a formalism that describes non-linear gravitational lensing by a circularly symmetric lens in the thin-lens approximation. This provides us with relatively simple expressions for first- and second-flexion in terms of only the surface density and projected mass distribution of the lens. We give details of exact lens models, in particular providing flexion calculations for a Sersic-law profile, which has become increasingly popular over recent years. We further provide a single resource for the analytic forms of convergence, shear, first- and second-flexion for the following mass distributions: a point mass, singular isothermal sphere (SIS); Navarro-Frenk-White (NFW) profile; Sersic-law profile. We quantitatively compare these mass distributions and show that the convergence and first-flexion are better indicators of the Sersic shape parameter, while for the concentration of NFW profiles the shear and second-flexion terms are preferred.