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 theories: 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 of four-point functions of two supercurrents with either R-currents. 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.