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The duality between the type IIB superstring theory in an $AdS_5 \times S^5$background with $N$ units of five-form flux and ${\cal N} =4$ super Yang--Millstheory with a $U(N)$ gauge group has been studied extensively. My version ofthe construction of the superstring world-sheet action is reviewed here. Thispaper is dedicated to Michael Duff on the occasion of his 70th birthday.

The $\mathbf{AdS_5 \times S^5}$ Superstring