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The $\omega$-power of a finitary language L over a finite alphabet $\Sigma$is the language of infinite words over $\Sigma$ defined by L $\infty$ := {w 0 w1. .. $\in$ $\Sigma$ $\omega$ | $\forall$i $\in$ $\omega$ w i $\in$ L}. The$\omega$-powers appear very naturally in Theoretical Computer Science in thecharacterization of several classes of languages of infinite words accepted byvarious kinds of automata, like B{\"u}chi automata or B{\"u}chi pushdownautomata. We survey some recent results about the links relating DescriptiveSet Theory and $\omega$-powers.

Descriptive Set Theory and $\omega$-Powers of Finitary Languages