  String theory in an exact plane wave background is explored. A new example of
singularity in the sense of string theory for nonsingular spacetime metric is
presented. The 4-tachyon scattering amplitude is constructed. The spectrum of
states found from the poles in the factorization turns out to be equivalent to
that of the theory in flat space-time. The massless vertex operator is obtained
from the residue of the first order pole.
