  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--Mills
theory with a $U(N)$ gauge group has been studied extensively. My version of
the construction of the superstring world-sheet action is reviewed here. This
paper is dedicated to Michael Duff on the occasion of his 70th birthday.
