  For an upstream supersonic flow past a straight-sided cone in $\R^3$ whose
vertex angle is less than the critical angle, a transonic (supersonic-subsonic)
shock-front attached to the cone vertex can be formed in the flow. In this
paper we analyze the stability of transonic shock-fronts in three-dimensional
steady potential flow past a perturbed cone. We establish that the self-similar
transonic shock-front solution is conditionally stable in structure with
respect to the conical perturbation of the cone boundary and the upstream flow
in appropriate function spaces. In particular, it is proved that the slope of
the shock-front tends asymptotically to the slope of the unperturbed
self-similar shock-front downstream at infinity.
