We show that supersymmetry is anomalous in 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 either with 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.