This article comprehensively reviews recent developments and research on deep learning-based (DL-based) techniques for integrated sensing and communication (ISAC) systems. ISAC, which combines sensing and communication functionalities, is regarded as a key enabler for 6G and beyond networks, as many emerging applications, such as vehicular networks and industrial robotics, necessitate both sensing and communication capabilities for effective operation. A unified platform that provides both functions can reduce hardware complexity, alleviate frequency spectrum congestion, and improve energy efficiency. However, integrating these functionalities on the same hardware requires highly optimized signal processing and system design, introducing significant computational complexity when relying on conventional iterative or optimization-based techniques. As an alternative to conventional techniques, DL-based techniques offer efficient and near-optimal solutions with reduced computational complexity. Hence, such techniques are well-suited for operating under limited computational resources and low latency requirements in real-time systems. DL-based techniques can swiftly and effectively yield near-optimal solutions for a wide range of sophisticated ISAC-related tasks, including waveform design, channel estimation, sensing signal processing, data demodulation, and interference mitigation. Therefore, motivated by these advantages, recent studies have proposed various DL-based approaches for ISAC system design. After briefly introducing DL architectures and ISAC fundamentals, this survey presents a comprehensive and categorized review of state-of-the-art DL-based techniques for ISAC, highlights their key advantages and major challenges, and outlines potential directions for future research.