Building controllers for legged robots with agility and intelligence has been one of the typical challenges in the pursuit of artificial intelligence (AI). As an important part of the AI field, deep reinforcement learning (DRL) can realize sequential decision making without physical modeling through end-to-end learning and has achieved a series of major breakthroughs in quadrupedal locomotion research. In this review article, we systematically organize and summarize relevant important literature, covering DRL algorithms from problem setting to advanced learning methods. These algorithms alleviate the specific problems encountered in the practical application of robots to a certain extent. We first elaborate on the general development trend in this field from several aspects, such as the DRL algorithms, simulation environments, and hardware platforms. Moreover, core components in the algorithm design, such as state and action spaces, reward functions, and solutions to reality gap problems, are highlighted and summarized. We further discuss open problems and propose promising future research directions to discover new areas of research.