Joint communication and radar sensing (JCR) represents an emerging research field aiming to integrate the above two functionalities into a single system, by sharing the majority of hardware, signal processing modules and, in a typical case, the transmitted signal. The close cooperation of the communication and sensing functions can enable significant improvement of spectrum efficiency, reduction of device size, cost and power consumption, and improvement of performance of both functions. Advanced signal processing techniques are critical for making the integration efficient, from transmission signal design to receiver processing. This paper provides a comprehensive overview of the state-of-the-art on JCR systems from the signal processing perspective. A balanced coverage on both transmitter and receiver is provided for three types of JCR systems, namely, communicationcentric, radar-centric, and joint design and optimization.