Autoregressive convolutional neural networks (CNNs) have been widely exploited for sequence generation tasks such as audio synthesis, language modeling and neural machine translation. WaveNet is a deep autoregressive CNN composed of several stacked layers of dilated convolution that is used for sequence generation. While WaveNet produces state-of-the art audio generation results, the naive inference implementation is quite slow; it takes a few minutes to generate just one second of audio on a high-end GPU. In this work, we develop the first accelerator platform~\textit{FastWave} for autoregressive convolutional neural networks, and address the associated design challenges. We design the Fast-Wavenet inference model in Vivado HLS and perform a wide range of optimizations including fixed-point implementation, array partitioning and pipelining. Our model uses a fully parameterized parallel architecture for fast matrix-vector multiplication that enables per-layer customized latency fine-tuning for further throughput improvement. Our experiments comparatively assess the trade-off between throughput and resource utilization for various optimizations. Our best WaveNet design on the Xilinx XCVU13P FPGA that uses only on-chip memory, achieves 66 faster generation speed compared to CPU implementation and 11 faster generation speed than GPU implementation.