  We elaborate a theory of giant vortices [1] based on an asymptotic expansion
in inverse powers of their winding number $n$. The theory is applied to the
analysis of vortex solutions in the abelian Higgs (Ginzburg-Landau) model.
Specific properties of the giant vortices for charged and neutral scalar fields
as well as different integrable limits of the scalar self-coupling are
discussed. Asymptotic results and the finite-$n$ corrections to the vortex
solutions are derived in analytic form and the convergence region of the
expansion is determined.
