The rapid growth of the Internet of Things (IoTs) has resulted in an explosive increase in data, and thus has raised new challenges for data processing units. Edge computing, which settles signal processing and computing tasks at the edge of networks rather than uploading data to the cloud, can reduce the amount of data for transmission and is a promising solution to address the challenges. One of the potential candidates for edge computing is a memristor, an emerging nonvolatile memory device that has the capability of in-memory computing. In this article, from the perspective of edge computing, we review recent progress on memristor-based signal processing methods, especially on the aspects of signal preprocessing and feature extraction. Then, we describe memristor-based signal classification and regression, and end-to-end signal processing. In all these applications, memristors serve as critical accelerators to greatly improve the overall system performance, such as power efficiency and processing speed. Finally, we discuss existing challenges and future outlooks for memristor-based signal processing systems.