In this paper, we firstly overview the application scenarios and the research progress in the area of communication and radar spectrum sharing (CRSS). We then propose a novel transceiver architecture and frame structure for a dual-functional radar-communication (DFRC) base station (BS) operating in the millimeter wave (mmWave) band, using the hybrid analog-digital (HAD) beamforming technique. We assume that the BS is serving a multi-antenna aided user equipment (UE) operating in a mmWave channel, which in the meantime actively detects multiple targets. Note that part of the targets also play the role of scatterers for the communication signal. Given this framework, we propose a novel scheme for joint target search and communication channel estimation relying on the omni-directional pilot signals generated by the HAD structure. Given a fully-digital communication precoder and a desired radar transmit beampattern, we propose to design the analog and digital precoders under non-convex constant-modulus (CM) and power constraints, such that the BS can formulate narrow beams towards all the targets, while pre-equalizing the impact of the communication channel. Furthermore, we design an HAD receiver that can simultaneously process signals from the UE and echo waves from the targets. By tracking the angular variation of the targets, we show that it is possible to recover the target echoes and mitigate the potential interference imposed on the UE signals by invoking the successive interference cancellation (SIC) technique, even when the radar and communication signals share the equivalent signal-to-noise ratio (SNR). The feasibility and the efficiency of the proposed approaches in realizing DFRC are verified via numerical simulations. Finally, our discussions are summarized by overviewing the open problems in the research field of CRSS.