In this article, an overview of advanced convex optimization approaches to multisensor beamforming is presented, and connections are drawn between different types of optimization-based beamformers that apply to a broad class of receive, transmit, and network beamformer design problems. It is demonstrated that convex optimization provides an indispensable set of tools for beamforming, enabling rigorous formulation and effective solution of both long-standing and emerging design problems.