Hydrogen energy is regarded as an ideal solution for addressing climate change issues and an indispensable part of future integrated energy systems. The most environmentally friendly hydrogen production method remains water electrolysis, where the electrolyzer constructs the physical interface between electrical energy and hydrogen energy. However, few articles have reviewed the electrolyzer from the perspective of power supply topology and control. This review is the first to discuss the positioning of the electrolyzer power supply in the future integrated energy system. The electrolyzer is reviewed from the perspective of the electrolysis method, the market, and the electrical interface modelling, reflecting the requirement of the electrolyzer for power supply. Various electrolyzer power supply topologies are studied and reviewed. Although the most widely used topology in the current hydrogen production industry is still single-stage AC/DC, the interleaved parallel LLC topology constructed by wideband gap power semiconductors and controlled by the zero-voltage switching algorithm has broad application prospects because of its advantages of high power density, high efficiency, fault tolerance, and low current ripple. Taking into account the development trend of the EL power supply, a hierarchical control framework is proposed as it can manage the operation performance of the power supply itself, the electrolyzer, the hydrogen energy domain, and the entire integrated energy system.