  The superstring in $D\!=\!3,4$ and 6 is invariant under an $N\!=\!D\!-\!2$
superconformal algebra based on $\widehat{S^{D-3}}$. There is a direct
relationship between this (world-sheet) symmetry and the super-Poincar\'e
(target space) symmetry. We establish this relationship using the light-cone
gauge, show how the statement generalizes to $D\!=\!10$ and examine the
properties of the $N\!=\!8$ superconformal algebra and the possible
implications of its existence.
