"The applications for multimedia services over high-speed networks increase dramatically in recent years. More and more developments are focused on the issues of IP-based and broadband wireline and wireless networks. Users need stable transmission quality when accessing the high-speed wired or wireless networks. In order to support a diverse of multimedia applications, the next-generation broadband networks have been required to satisfy the Quality of Service (QoS) requirements. Real-time and precise traffic control and scheduling mechanisms are essential to achieve the QoS guarantee. Therefore, the following topics are studied. The group of mathematical theory of networks have two subprojects named “Nonblocking multistage networks” and “Mathematical Theory for Cellular Neural Networks (CNN)” leaded by Frank K. Hwang and Song-Sun Lin respectively. We give a brief description.