In this paper, stability analysis and decentralized control problems are addressed for linear and sector-nonlinear complex dynamical networks. Necessary and sufficient conditions for stability and stabilizability under a special decentralized control strategy are given for linear networks. Especially, two types of linear regular networks, star-shaped networks and globally coupled networks, are studied in detail. A dynamical network is viewed as a large-scale system composing of some subsystems, based on which the relationship between the stability of a network and the stability of its corresponding subsystems is investigated. It is pointed out that some subsystems must be unstable for the whole network to be stable in some special cases. Moreover, a controller design method based on a parameter-dependent Lyapunov function is provided. Furthermore, interconnected Lur’e systems and symmetrical networks of Lur’e systems are similarly studied. The test of absolute stability of a network of Lur’e systems is separated into the test of absolute stability of several independent Lur’e systems. Finally, several numerical examples are given to illustrate the theoretical results.