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After the invention of a new mechanical element called "inerter" in 2002, research interest in passive network synthesis has been revived and this field has again become active and essential. The unique compendium highlights the synthesis of passive electrical or mechanical networks, which is motivated by the vibration control based on a new type of mechanical elements named inerter. It introduces important fundamental concepts of passive network synthesis, and presents recent results on this topic. These new results concern mainly the economical realizations of low-degree functions as RLC networks (damper-spring-inerter networks), the synthesis of n-port resistive networks, and the synthesis of low-complexity mechanical networks. They can be directly applied to the optimization and design of various inerter-based mechanical control systems, such as suspension systems, vibration absorbers, building vibration systems, etc. This useful reference text provides important methodologies and results for researchers in the fields of circuit theory, vibration system control, passive systems, control theory, and electrical engineering.