  We show that supersymmetry is anomalous in ${\cal N}=1$ superconformal
quantum field theories (SCFTs) with an anomalous R-symmetry. This anomaly was
originally found in holographic SCFTs at strong coupling. Here we show that
this anomaly is present in general and demonstrate it for the massless
superconformal Wess-Zumino model via a one loop computation. The anomaly
appears first in four-point functions of two supercurrents with either two
R-currents or with an R-current and an energy momentum tensor. In fact, the
Wess-Zumino consistency conditions together with the standard R-symmetry
anomaly imply the existence of the anomaly. We outline the implications of this
anomaly.
