High-quality datasets are typically required for accomplishing data-driven tasks, such as training medical diagnosis models, predicting real-time traffic conditions, or conducting experiments to validate research hypotheses. Consequently, open dataset search, which aims to ensure the efficient and accurate fulfillment of users' dataset requirements, has emerged as a critical research challenge and has attracted widespread interest. Recent studies have made notable progress in enhancing the flexibility and intelligence of open dataset search, and large language models (LLMs) have demonstrated strong potential in addressing long-standing challenges in this area. Therefore, a systematic and comprehensive review of the open dataset search problem is essential, detailing the current state of research and exploring future directions. In this survey, we focus on recent advances in open dataset search beyond traditional approaches that rely on metadata and keywords. From the perspective of dataset modalities, we place particular emphasis on example-based dataset search, advanced similarity measurement techniques based on dataset content, and efficient search acceleration techniques. In addition, we emphasize the mutually beneficial relationship between LLMs and open dataset search. On the one hand, LLMs help address complex challenges in query understanding, semantic modeling, and interactive guidance within open dataset search. In turn, advances in dataset search can support LLMs by enabling more effective integration into retrieval-augmented generation (RAG) frameworks and data selection processes, thereby enhancing downstream task performance. Finally, we summarize open research problems and outline promising directions for future work. This work aims to offer a structured reference for researchers and practitioners in the field of open dataset search.