Challenged by urbanization and increasing travel needs, existing transportation systems call for new mobility paradigms. In this article, we present the fleet management and charging scheduling of a shared mobility-on-demand system, whereby electric vehicle fleets are operated by a centralized platform to provide customers with mobility service. We provide a comprehensive review of system operation based on the operational objectives. The fleet scheduling strategies are categorized into four types: i) order dispatching, ii) order-dispatching and rebalancing, iii) order-dispatching, rebalancing and charging, and iv) extended. Specifically, we first identify mathematical modeling techniques implemented in the transportation network, then analyze and summarize the solution approaches including mathematical programming, reinforcement learning, and hybrid methods. The advantages and disadvantages of different models and solution approaches are compared. Finally, we present research outlook in various directions.