  We discuss vortex solutions of the abelian Higgs model in the limit of large
winding number $n$. We suggest a framework where a topological quantum number
$n$ is associated with a ratio of dynamical scales and a systematic expansion
in inverse powers of $n$ is then derived in the spirit of effective field
theory. The general asymptotic form of ${\it giant}$ vortices is obtained. For
critical coupling the axially symmetric vortices become ${\it integrable}$ in
the large-$n$ limit and we present the corresponding analytic solution. The
method provides simple asymptotic formulae for the vortex shape and parameters
with accuracy that can be systematically improved, and can be applied to
topological solitons of other models. After including the next-to-leading terms
the approximation works remarkably well down to $n=1$.
