Integrated sensing and communication (ISAC) has been widely recognized as a key technology in future sixth-generation (6G) wireless networks, especially for emerging applications and scenarios demanding both high-performance sensing and communication functionalities. The ISAC technique has the potential of enabling sensing and communication functionalities in a single hardware platform, even sharing the same waveform, thus, achieving the reduced cost of hardware implementation and the efficient use of spectrum resources. In this paper, we commence with the discussion on sensing and communication at the current stage from the view of motivations, applications, and challenges. Then, we provide a comprehensive survey of the state-of-the-art approaches to the ISAC technique from the waveform design perspective. To be specific, we classify the waveform design methods into three categories, namely, communication-centric waveform design, sensing-centric waveform design, and joint waveform optimization and design. In addition, potential research directions and challenges for future ISAC waveform design are outlined.