As a clean and renewable energy source for sustainable development, hydrogen energy has gained a lot of attention from the general public and researchers. Hydrogen production by electrolysis of water is the most important approach to producing hydrogen, and it is also the main way to realize carbon neutrality. In this paper, the main technologies of hydrogen production by electrolysis of water are discussed in detail; their characteristics, advantages, and disadvantages are analyzed; and the selection criteria and design criteria of catalysts are presented. The catalysts used in various hydrogen production technologies and their characteristics are emphatically expounded, aiming at optimizing the existing catalyst system and developing new high-performance, high-stability, and low-cost catalysts. Finally, the problems and solutions in the practical design of catalysts are discussed and explored.