This paper presents a novel transceiver architecture and frame structure for a dual-functional radar-communication (DFRC) base station (BS) operating in the millimeter wave (mmWave) band. The BS serves a multi-antenna user equipment (UE) while detecting multiple targets, some of which act as scatterers for the communication signal. The proposed system employs a hybrid analog-digital (HAD) beamforming technique to reduce the number of RF chains and enable efficient joint radar and communication operations. The DFRC frame structure is designed to comply with state-of-the-art time-division duplex (TDD) protocols and is divided into three stages: 1) radar target search and communication channel estimation, 2) radar transmit beamforming and downlink communication, and 3) radar target tracking and uplink communication. In each stage, joint signal processing approaches are proposed to fulfill both target detection and communication tasks. In Stage 1, the angles of arrival (AoAs) of all potential targets and communication channel parameters are estimated using both downlink (DL) and uplink (UL) pilots. In Stage 2, a novel joint HAD transmit beamforming design is proposed to formulate directional beams towards the angles of interest while equalizing the communication channel. In Stage 3, the angular variation of the targets is tracked by simultaneously processing the echoes of the targets while decoding the UL signal transmitted from the UE. The proposed system is evaluated through numerical simulations, which demonstrate its feasibility and efficiency in realizing DFRC. The paper also discusses the open problems in the research field of communication and radar spectrum sharing (CRSS).This paper presents a novel transceiver architecture and frame structure for a dual-functional radar-communication (DFRC) base station (BS) operating in the millimeter wave (mmWave) band. The BS serves a multi-antenna user equipment (UE) while detecting multiple targets, some of which act as scatterers for the communication signal. The proposed system employs a hybrid analog-digital (HAD) beamforming technique to reduce the number of RF chains and enable efficient joint radar and communication operations. The DFRC frame structure is designed to comply with state-of-the-art time-division duplex (TDD) protocols and is divided into three stages: 1) radar target search and communication channel estimation, 2) radar transmit beamforming and downlink communication, and 3) radar target tracking and uplink communication. In each stage, joint signal processing approaches are proposed to fulfill both target detection and communication tasks. In Stage 1, the angles of arrival (AoAs) of all potential targets and communication channel parameters are estimated using both downlink (DL) and uplink (UL) pilots. In Stage 2, a novel joint HAD transmit beamforming design is proposed to formulate directional beams towards the angles of interest while equalizing the communication channel. In Stage 3, the angular variation of the targets is tracked by simultaneously processing the echoes of the targets while decoding the UL signal transmitted from the UE. The proposed system is evaluated through numerical simulations, which demonstrate its feasibility and efficiency in realizing DFRC. The paper also discusses the open problems in the research field of communication and radar spectrum sharing (CRSS).