This paper provides a comprehensive overview of the state-of-the-art related to the implementation of battery electric buses (BEBs) in cities. In recent years, bus operators have started focusing on the electrification of their fleet to reduce the air pollutants in cities, which has led to a growing interest from the scientific community. This paper presents an analysis of the BEB powertrain topology and the charging technology of BEBs, with a particular emphasis on the power electronics systems. Moreover, the different key technical requirements to facilitate the operation of BEBs are addressed. Accordingly, an in-depth review on vehicle scheduling, charger location optimization and charging management strategies is carried out. The main findings concerning these research fields are summarized and discussed. Furthermore, potential challenges and required further developments are determined. Based on this analysis, it can be concluded that an accurate energy consumption assessment of their BEBs is a must for bus operators, that real-time, multi-objective smart charging management strategies with V2X features should be included when performing large bus fleet scheduling and that synchronized opportunity charging, smart green depot charging, and electric bus rapid transit can further reduce the impact on the grid. This review paper should help to enable a smarter and more efficient integration of BEBs in cities in the future.