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The design of a highly configurable continuous flow mixed-radix (CFMR) fast Fourier transform (FFT) processor is presented. It computes fixed-point complex FFTs and inverse FFTs (IFFTs), and utilizes a flexible addressing scheme to enable runtime configuration of the FFT length from 16-points to 4096-points. A configurable block floating point (BFP) unit increases numerical performance. Compared to a floating point Matlab FFT function, the accuracy of the proposed architecture is 80 dB for a 64-point FFT and 74 dB for a 1024-point FFT with random complex input data.