Vortex rings are three-dimensional flows for which the vorticity distribution is concentrated in a solid torus. A natural way to construct such flows is to consider axisymmetric solutions without swirl of the 3D Euler or Navier-Stokes equations, assuming that the initial data are suitably localized. We focus on the particular case where the initial vorticity is just one circular vortex filament, the circulation of which may be arbitrarily large. It is known that the axisymmetric Navier-Stokes equations are globally well-posed in that case, and our goal is to study such viscous vortex rings in the vanishing viscosity limit. In this talk, we perform a perturbation expansion of the solution, and we use it in particular to determine the translation speed of the vortex ring beyond the binormal flow approximation. This is based on joint work with Vladimir Sverak.