  We show how realization theory can be used to find the solutions of the
Carath\'eodory extremal problem on the symmetrized bidisc \[ G
\stackrel{\rm{def}}{=} \{(z+w,zw):|z|<1, \, |w|<1\}. \] We show that,
generically, solutions are unique up to composition with automorphisms of the
disc. We also obtain formulae for large classes of extremal functions for the
Carath\'eodory problems for tangents of non-generic types.
