Abstract: Low-Density Parity-Check (LDPC) codes have become a popular forward error correction scheme owing to their theoretical performance approaching the Shannon Limit. However, LDPC codes exhibit an error floor in the region of high signal-to-noise ratio for most iterative decoding algorithms. As for Additive White Gaussian Noise (AWGN) channels, the error floor of LDPC codes is primarily caused by special topologies in the corresponding Tanner graph, which lead to decoding failure known as trapping sets. The existence of the trapping sets makes it difficult for the decoder to detect the errors when the nodes in these sets have data errors, which significantly increases the probability of decoding failure. This characteristic limits the use of LDPC codes in applications that require low target error rates. Therefore, research and insights to the trapping sets are of great significance for addressing the error floor problem. In this paper, recent research on trapping sets in LDPC codes is reviewed, and the basic definition, effect mechanism, searching methods, and elimination methods are systematically introduced. Finally, some unsolved problems, challenges and future development prospects on trapping sets and error floor are further discussed.