Future wireless communication systems will evolve toward multi-functional integrated systems to improve spectrum utilization and reduce equipment sizes. A joint radar and communication (JRC) system, which can support simultaneous information transmission and target detection, has been regarded as a promising solution for emerging applications such as autonomous vehicles. In JRC, data security and privacy protection are critical issues. Thus, we first apply covert communication into JRC and propose a joint radar and covert communication (JRCC) system to achieve high spectrum utilization and secure data transmission simultaneously. In the JRCC system, an existence of sensitive data transmission is hidden from a maliciously observant warden. However, the performance of JRCC is restricted by severe signal propagation environment and hardware devices. Fortunately, reconfigurable intelligent surfaces (RISs) can change the signal propagation smartly to improve the networks performance with low cost. We first overview fundamental concepts of JRCC and RIS and then propose the RIS-aided JRCC system design. Furthermore, both covert communication and radar performance metrics are investigated and a game theory-based covert rate optimization scheme is designed to achieve secure communication. Finally, we present several promising applications and future directions of RIS-aided JRCC systems.