The population of electric vehicles (EVs) has grown rapidly over the past decade due to the development of EV technologies, battery materials, charger facilities, and public charging services. Many governments have implemented plans to ban fossil fuel vehicles considering the significance of EVs in reducing greenhouse gas emissions. However, due to the battery material characteristics and charger power limitations, the EV charging process requires more time than is needed to fill a non-EV with fuel at a gasoline station, causing drivers to experience range anxiety and impeding the promotion of EVs. Hence, the battery swapping station (BSS) model has been proposed as an alternative method. Recently, researchers have studied the BSS approach by proposing various operation systems and optimization methods, and BSS service operators have successfully implemented swapping at commercial and private stations. This paper reviews the state-of-the-art BSS literature and business models, where the BSS offers a recharged battery to an incoming EV with a low state-of-charge. First, four operation modes are presented: a single BSS, multiple BSSs, an integrated BSS and battery charging station (BCS), and multiple BSSs and BCSs. Then, the BSS decision scenarios are surveyed in relation to five operational areas, i.e., charging schedule, service policy, construction and planning, dispatching and routing, and power management, where the scenarios are compared in terms of the BSS mode, decision maker, EV category, number of battery types, vehicle to grid, and focus and objective. Finally, the survey concludes with a discussion of several future research directions for EV BSSs.