To further improve the system capacity for 5G, we explore the integration of non-orthogonal multiple access (NOMA) in mmWave communications (mmWave-NOMA) for future 5G systems. Compared with the conventional NOMA, the distinguishing feature of mmWave-NOMA is that, it is usually characterized by transmit/receive beamforming with large antenna arrays. In this paper, we focus on the design challenges of mmWave-NOMA due to beamforming. Firstly, we study how beamforming affects the sum-rate performance of mmWave-NOMA, and find that with conventional single-beam forming, the performance may be offset by the relative angle between NOMA users. Then, we consider multi-beam forming for mmWave-NOMA, which is shown to be able to achieve promising performance enhancement as well as robustness. We further investigate the challenging joint design of the intertwined power allocation and user pairing for mmWave-NOMA. We also discuss the challenges and propose some potential solutions in detail. Finally, we consider hybrid spatial division multiple access (SDMA) and NOMA in mmWave communications, where some possible system configurations and the corresponding solutions are discussed to address the multi-user issues including multi-user precoding and multi-user interference (MUI) mitigation.