The detection and correction of insertion/deletion (indel) errors have become increasingly critical in domains such as traditional mobile communication systems, the Internet of Things (IoT), smart homes, smart healthcare, vehicular networks, and large-scale urban infrastructure, establishing it as a prominent research focus. As a typical form of synchronization error, the randomness and asymmetry of indel errors severely disrupt symbol alignment and induce significant synchronization drift, thereby imposing substantial challenges on reliable data transmission. This paper systematically reviews methodologies for detecting and correcting indel errors, tracing their evolution from model-driven to data-driven paradigms. First, we summarize the traditional technical framework, which includes synchronization markers, edit distance (ED) codes, sequence alignment, trellis/convolutional structures, and probabilistic models, with an analysis of their theoretical foundations, representative algorithms, and applicable scenarios. Next, we focus on recent advances in deep learning (DL)-based synchronization recovery methods and semantic communication-driven intelligent error correction frameworks, highlighting their distinct advantages over conventional approaches in handling complex channels and unstructured data. Finally, we outline the current research landscape and key challenges in this field and propose future directions for emerging scenarios such as 6th Generation (6G) ultra-reliable communication, satellite links, and ultra-high-density storage. This review aims to provide comprehensive insights and guidance for the design of synchronization and error correction mechanisms in next-generation communication systems.