The scattering of plane surface water waves by both surface-piercing and submerged axisymmetric bed formations has been investigated by a number of authors. We extend previous contributions in two ways: we employ the recently derived modified mild-slope equation to approximate the fluid motion and we consider arbitrary axisymmetric topography, subject only to the restriction imposed by the mild-slope approximation. We thereby derive a robust method for calculating the scattered wave field for the small seabed slopes typical of most real situations and make a contribution to the phenomena of near-resonance and near-trapping over submerged axisymmetric shoals, which have previously been detected for only one idealized bedform.