  As is well-known, numerical experiments show that Napoleon's Theorem for
planar triangles does not extend to a similar statement for triangles on the
unit sphere $S^2$. Spherical triangles for which an extension of Napoleon's
Theorem holds are called ``Napoleonic'', and until now the only known examples
have been equilateral. In this paper we determine all Napoleonic spherical
triangles, including a class corresponding to points on a 2-dimensional
ellipsoid, whose Napoleonisations are all congruent. Other new classes of
examples are also found, according to different versions of Napoleon's Theorem
for the sphere. The classification follows from successive simplifications of a
complicated original algebraic condition, exploiting geometric symmetries and
algebraic factorisations.
