This paper reviews the burgeoning field of data-driven algorithms and their application in solving increasingly complex decision-making, optimization, and control problems within active distribution networks. By summarizing a wide array of use cases, including network reconfiguration and restoration, crew dispatch, Volt-Var control, dispatch of distributed energy resources, and optimal power flow, we underscore the versatility and potential of data-driven approaches to improve active distribution system operations. The categorization of these algorithms into four main groups-mathematical optimization, end-to-end learning, learning-assisted optimization, and physics-informed learning-provides a structured overview of the current state of research in this domain. Additionally, we delve into enhanced algorithmic strategies such as non-centralized methods, robust and stochastic methods, and online learning, which represent significant advancements in addressing the unique challenges of active distribution systems. The discussion extends to the critical role of datasets and test systems in fostering an open and collaborative research environment, essential for the validation and benchmarking of novel data-driven solutions. In conclusion, we outline the primary challenges that must be navigated to bridge the gap between theoretical research and practical implementation, alongside the opportunities that lie ahead. These insights aim to pave the way for the development of more resilient, efficient, and adaptive active distribution networks, leveraging the full spectrum of data-driven algorithmic innovations.