  The established results concerning the BRS cohomology of supersymmetric
theories in four space-time dimensions are briefly reviewed. The current status
of knowledge concerning supersymmetry anomalies and the possibility that
supersymmetry breaks itself through anomalies in local composite operators is
then discussed.
  It turns out that the simplest allowable supersymmetry anomalies occur only
in conjunction with the spontaneous breaking of gauge symmetry. A simple
example of such a possible supersymmetry anomaly is presented.
