This paper presents a linear matrix inequality (LMI) approach to the multiobjective synthesis of linear output-feedback controllers. The design objectives can include $H_{\infty}$ performance, $H_2$ performance, passivity, asymptotic disturbance rejection, time-domain constraints, and constraints on the closed-loop pole location. These objectives can be specified on different channels of the closed-loop system. By formulating all objectives in terms of a common Lyapunov function, controller design reduces to solving a system of linear matrix inequalities. The paper provides a comprehensive overview of the design technique, including the problem statement, various specifications, and the methodology for transforming analysis results into synthesis LMI constraints. It also discusses the benefits and limitations of the Lyapunov shaping paradigm and the linearizing change of variable used to linearize the problem. The paper concludes with a catalog of LMI constraints for full-order synthesis, providing a practical guide for implementing the proposed approach.This paper presents a linear matrix inequality (LMI) approach to the multiobjective synthesis of linear output-feedback controllers. The design objectives can include $H_{\infty}$ performance, $H_2$ performance, passivity, asymptotic disturbance rejection, time-domain constraints, and constraints on the closed-loop pole location. These objectives can be specified on different channels of the closed-loop system. By formulating all objectives in terms of a common Lyapunov function, controller design reduces to solving a system of linear matrix inequalities. The paper provides a comprehensive overview of the design technique, including the problem statement, various specifications, and the methodology for transforming analysis results into synthesis LMI constraints. It also discusses the benefits and limitations of the Lyapunov shaping paradigm and the linearizing change of variable used to linearize the problem. The paper concludes with a catalog of LMI constraints for full-order synthesis, providing a practical guide for implementing the proposed approach.