This paper provides a comprehensive survey on recent advances in deep learning (DL) techniques for the channel coding problems. Inspired by the recent successes of DL in a variety of research domains, its applications to the physical layer technologies have been extensively studied in recent years, and are expected to be a potential breakthrough in supporting the emerging use cases of the next generation wireless communication systems such as 6G. In this paper, we focus exclusively on the channel coding problems and review existing approaches that incorporate advanced DL techniques into code design and channel decoding. After briefly introducing the background of recent DL techniques, we categorize and summarize a variety of approaches, including model-free and mode-based DL, for the design and decoding of modern error-correcting codes, such as low-density parity check (LDPC) codes and polar codes, to highlight their potential advantages and challenges. Finally, the paper concludes with a discussion of open issues and future research directions in channel coding.