Automatic discovery of community structures in complex networks is a fundamental task in many disciplines, including social science, engineering, and biology. A quantitative measure called modularity (Q) has been proposed to effectively assess the quality of community structures. Several community discovery algorithms have since been developed based on the optimization of Q. However, this optimization problem is NP-hard, and the existing algorithms have a low accuracy or are computationally expensive. In this paper, we present an efficient spectral algorithm for modularity optimization. When tested on a large number of synthetic or real-world networks, and compared to the existing algorithms, our method is efficient and and has a high accuracy. In addition, we have successfully applied our algorithm to detect interesting and meaningful community structures from real-world networks in different domains, including biology, medicine and social science. Due to space limitation, results of these applications are presented in a complete version of the paper available on our Website (http://cse .wustl.edu/ <sup>~</sup>jruan/).