Device-to-device communications enable two proximity users to transmit signal directly without going through the base station. It can increase network spectral efficiency and energy efficiency, reduce transmission delay, offload traffic for the BS, and alleviate congestion in the cellular core networks. However, many technical challenges need to be addressed for D2D communications to harvest the potential benefits, including device discovery and D2D session setup, D2D resource allocation to guarantee QoS, D2D MIMO transmission, as well as D2D-aided BS deployment in heterogeneous networks. In this article, the basic concepts of D2D communications are first introduced, and then existing fundamental works on D2D communications are discussed. In addition, some potential research topics and challenges are also identified.