  We study the spectrum of asymptotic states in the spin-chain description of
planar N=4 SUSY Yang-Mills. In addition to elementary magnons, the asymptotic
spectrum includes an infinite tower of multi-magnon bound states with exact
dispersion relation, Delta-J_{1} = sqrt{Q^{2}+(lambda/pi^2)sin^2(p/2)}, where
the positive integer Q is the number of constituent magnons. These states
account precisely for the known poles in the exact S-matrix. Like the
elementary magnon, they transform in small representations of supersymmetry and
are present for all values of the 't Hooft coupling. At strong coupling we
identify the dual states in semiclassical string theory.
