Synergistic design of communications and radar systems with common spectral and hardware resources is heralding a new era of efficiently utilizing a limited radio-frequency (RF) spectrum. Such a joint radar communications (JRC) model has advantages of low cost, compact size, less power consumption, spectrum sharing, improved performance, and safety due to enhanced information sharing. Today, millimeter-wave (mmwave) communications have emerged as the preferred technology for short distance wireless links because they provide transmission bandwidth that is several gigahertz wide. This band is also promising for short-range radar applications, which benefit from the high-range resolution arising from large transmit signal bandwidths. Signal processing techniques are critical to the implementation of mm-wave JRC systems. Major challenges are joint waveform design and performance criteria that would optimally trade off between communications and radar functionalities. Novel multiple-input, multiple-output (MIMO) signal processing techniques are required because mm-wave JRC systems employ large antenna arrays. There are opportunities to exploit recent advances in cognition, compressed sensing, and machine learning to reduce required resources and dynamically allocate them with low overheads. This article provides a signal processing perspective of mm-wave JRC systems with an emphasis on waveform design.