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To provide service to an abundant number of communication users and to avoid the spectrum scarcity problem, many researchers are fascinated to work towards the convergence of radar sensing and communication systems. In addition, future intelligent systems like autonomous vehicles, Vehicle-to-everything (V2X), Unmanned Aerial Vehicles (UAV), and all smart systems are going to implement both radar and communication systems on the same platform, which motivates the researchers to focus on the development of Joint Radar-Communication Systems (JRCS). Cooperative Radar-Communication System (CRCS) and Dual Functional Radar Communication (DFRC) systems provide an opportunity for communication users to utilize radar resources without disturbing radar operation. Waveform design is essential in the development of new models and designs related to joint radar-sensing and communication systems. A cooperative radar communication system uses separate waveforms for radar and communication systems. The DFRC system uses the same waveform for radar and communication operations. So to model both joint radar communication systems one should have a clear idea regarding waveform design and its approaches. Therefore, this review paper focused on different waveform design approaches for modeling CRCS and DFRC systems. In addition, the prime objective of this review paper is to give a detailed view of the existing cooperative and dual-function waveform design approaches and provide a kick-start for new learners to work on this area.