Joint communication and radar sensing (JCR) represents an emerging research field aiming to integrate the above two functionalities into a single system, sharing a majority of hardware and signal processing modules and, in a typical case, sharing a single transmitted signal. It is recognised as a key approach in significantly improving spectrum efficiency, reducing device size, cost and power consumption, and improving performance thanks to potential close cooperation of the two 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 JCR systems from the signal processing perspective, with a focus on state-of-the-art. A balanced coverage on both transmitter and receiver is provided for three types of JCR systems, communication-centric, radar-centric, and joint design and optimization.