  We prove the existence of rational maps whose Julia sets are Sierpi\'{n}ski
carpets having positive area. Such rational maps can be constructed such that
they either contain a Cremer fixed point, a Siegel disk or are infinitely
renormalizable. We also construct some Sierpi\'{n}ski carpet Julia sets with
zero area but with Hausdorff dimension two. Moreover, for any given number
$s\in(1,2)$, we prove the existence of Sierpi\'{n}ski carpet Julia sets having
Hausdorff dimension exactly $s$.
