A cylindrical column of liquid will tend to disintegrate into droplets as a result of the Rayleigh capillary instability associated with surface tension. Suppose, however, that the column is composed of a ferrofluid, a liquid with tiny magnetisable particles in suspension which behaves effectively like a liquid magnet. If the column is exposed to a magnetic field (with concentric field lines) of sufficient strength, it can be fully stabilised. Such a system then offers a venue for studying axisymmetric nonlinear waves, including solitary waves. The latter are expected to exist based on theoretical predictions using weakly nonlinear analysis; in fact, the KdV equation with the famous sech^2-shaped soliton solution can be shown to hold. Waves of this type were observed experimentally for the first time by Bourdin et al in France in 2010. In this talk we present fully nonlinear results for solitary waves propagating on a column of ferrofluid. We show the existence of static solitary waves, and propagating waves of elevation and depression. Our findings demonstrate the need to re-evaluate the experimental observations of Bourdin et al.