© 2015 Chinese Physical Society. The model, structure and dynamics of complex systems and networks are studied to control complex systems, which reflects the ability to understanding complex systems. Recently, the research on controllability of complex networks by using control theory and complexity science has attracted much attention. It has been investigated extensively by many scientists from various fields, and many meaningful achievements have been obtained in the past few years. In this paper, the process of controllability of complex networks is discussed, the framework of structural controllability based on maximum matching is introduced in detail, and the relevant research status since 2011 is summarized. Controllabilities of complex networks are introduced in the following aspects: different types of controllabilities, relationship between controllability and network statistical characteristics, classification and measures based on controllability, robustness of controllability, and optimization methods of controllability. Finally, the questions urgent to solve in controllability are discussed, so as to give a help to the the study in this respect. There are five sections in this paper, which involve with different aspects of controllability. In the introduction section, the research work of controllability since 2011 is briefly mentioned, and the difference between controllability and previous pinning controllability is clarified. In the second section, the concept of controllability and different types of controllabilities are discussed in detail, including structural controllability, exact controllability, controllability with edge dynamics and controllability with nodal dynamics. In the third section, the relationship between controllability and network structure is investigated, especially the effects of common statistical characteristics and low-degree nodes on controllability. In the fourth section, the measures based on controllability are introduced, which includes control profiles, control range, control centrality, control capacity and control modality. In the fifth section, the research work about control robustness is discussed from robustness measures to optimization methods. In the fifth section, the optimization methods of controllability are introduced, which are classified into two different strategies: topology and edge direction.