The superstring in D = 3,4 and 6 is invariant under an N = D - 2 super
conformal algebra based on S(D-3). There is a direct relationship betw
een this (world-sheet) symmetry and the super-Poincare (target space)
symmetry. We establish this relationship using the light-cone gauge, s
how how the statement generalizes to D = 10 and examine the properties
of the N = 8 superconformal algebra and the possible implications of
its existence.