As the global financial landscape undergoes profound transformation, the blockchain technology has emerged as a cornerstone of Web 3.0 finance and a pivotal frontier in financial technology innovation. However, its decentralized and pseudonymous nature has also been exploited by malicious actors to circumvent regulatory oversight, facilitate money laundering, and conduct other illicit financial activities, posing substantial risks to both national and global financial security. Consequently, there is an urgent need to systematically assess the current progress in anti-money laundering (AML) research, anticipate future directions in blockchain-based AML technologies, and develop effective countermeasures to mitigate the evolving financial security challenges associated with blockchain applications. This study provides a comprehensive review of AML research in blockchain systems, examining the foundational AML frameworks, including traditional AML models and blockchain-based money laundering methodologies. It categorizes existing AML techniques into three primary approaches: rule-based methods, such as transaction parameter threshold setting, address-entity association analysis, and cross-chain association analysis; machine learning-based approaches, including support vector machines, logistic regression, decision trees, random forests, k-means clustering, and combining off-chain information; and deep learning-based methodologies, encompassing convolutional neural networks, recurrent neural networks, graph neural networks, and transformer-based models. Furthermore, this study discusses the practical applications of these techniques and reviews commonly used datasets that support AML research. Looking ahead, the advancement of AML technologies in blockchain systems necessitates progress in several critical areas: the construction of AML datasets capable of addressing data imbalance and annotation uncertainty, development of trusted AML algorithms, design of detection mechanisms for covert financial activities, and formulation of privacy-preserving yet regulation-compliant AML solutions. Strengthening these capabilities will enhance the effectiveness of AML frameworks within blockchain ecosystems and contribute to the secure and sustainable development of the digital economy.