This paper concerns the determination of axially-symmetric MHD equilibria with arbitrary cross-section. – A numerical solution to the appropriate 2-D free-boundary elliptic non-linear PDE has been used to relate current profile, coil configuration, and plasma geometry for non-circular tokamaks with external field-shaping coil systems. The bifurcation properties of the solution spaces for such tokamaks have been studied and the parameter regions wherein bifurcation occurs determined. The structural stability of the resulting equilibrium suggests ways of controlling the shape of different plasmas (in particular doublet plasmas) over a wide range of plasma parameters.