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In recent years, we have witnessed technical breakthroughs in a wide variety of topics where the key to success is the use of convex optimization. In fact, convex optimization has now emerged as a major signal processing tool that has made a significant impact on numerous problems previously considered intractable. Considering the foundational nature and potential impact of convex optimization in signal processing, we have put together this special issue that aims to provide tutorials of convex optimization techniques (including available software) and various successful signal processing applications. Our goal is not only to contribute to the diffusion of recent developments in this research area within the signal processing community, but also to spur further advances in and applications of convex optimization for signal processing.