Compared with omni-directional antennas, directional antennas have many merits, such as lower interference, better spatial reuse, longer transmission range, and improved network capacity. Directional antennas enable numerous emerging outdoor and indoor applications, which have been addressed in many recent studies. Despite the advances in wireless networks with directional antennas (DAWNs), there are many research challenges in all layers of DAWNs. This paper presents a detailed study on recent advances and open research issues on DAWNs. Firstly, we briefly introduce the classification of directional antennas, antenna radiation patterns, antenna modes, and the challenges in the physical layer of DAWNs. We then present research issues on the medium access control (MAC) layer, followed by the current solutions as well as open research problems on the MAC layer of DAWNs. In addition, we also discuss the research issues on the routing layer and the transport layer. Moreover, other research challenges on the performance evaluation of DAWNs and a brief introduction of indoor DAWNs are given in this paper as well. In conclusion, we summarize the current research issues on DAWNs as well as prospects in the future.