In contrast to the long-held belief that wireless systems can only work in half-duplex mode, full-duplex (FD) systems are able to concurrently transmit and receive information over the same frequency bands to theoretically enable a twofold increase in spectral efficiency. Despite their significant potential, FD systems suffer from an inherent self-interference (SI) due to a coupling of the transmit signal to its own FD receive chain. Selfinterference cancellation (SIC) techniques are the key enablers for realizing the FD operation, and they could be implemented in the propagation, analog, and/or digital domains. Particularly, digital domain cancellation is typically performed using modeldriven approaches, which have proven to be insufficient to seize the growing complexity of forthcoming communication systems. For the time being, machine learning (ML) datadriven approaches have been introduced for digital SIC to overcome the complexity hurdles of traditional methods. This paper reviews and summarizes the recent advances in applying ML to SIC in FD systems. Further, it analyzes the performance of various ML approaches using different performance metrics, such as the achieved SIC, training overhead, memory storage, and computational complexity. Finally, this paper discusses the challenges of applying ML-based techniques to SIC, highlights their potential solutions, and provides a guide for future research directions.