Recently, we have witnessed the advances of convex optimization, including theoretical aspects (e.g., interior-point methods) and general-purpose software tools (e.g., SeDuMi). It has been recognized as an extremely useful technique in helping with addressing a wide range of signal processing problems, including those previously considered intractable. This talk aims to provide a tutorial of convex optimization techniques relevant to signal processing, which could be used to identify and formulate convex optimization problems arising in signal processing scenarios, and characterize the structure of optimal solutions to signal processing problems. The talk will also cover some successful signal processing applications to concretely demonstrate the effectiveness of convex optimization and its impact on signal processing.