With the improvement of fountain clock reliability and the gradual maturity of laser cooling technology, the fountain clocks can not only serve as primary reference clocks to steer atomic time scales and improve their precision, but are also increasingly used to construct composite timekeeping clocks by steering hydrogen masers, playing an increasingly vital role in the time-frequency domain. Research on fountain clocks and continuous improvement of their performance indicators including frequency stability are of great practical value and scientific significance. As a secondary realization of the definition of the second, the ground-state hyperfine transition of rubidium atoms features a small collision cross section. Coupled with the high operational reliability of cooling lasers, rubidium fountain clocks possess promising application prospects and have gradually entered practical application. In terms of long-term precision, rubidium fountain clocks have achieved excellent performance. The daily stability is on the order of 10-16; the half-month and monthly stability correspond to small coefficients of 10-16, and the rubidium fountain clock developed in Russia stands out with the best performance, reaching large coefficients of 10-17. The annual stability reaches 10-17 of small coefficients; the accuracy is on the order of 10-16, and the drift rate reaches 10-18 per day of small coefficients. Domestic and international research institutions have distinct emphases on laser cooling and optics, microwave cavity and Ramsey interrogation, frequency-shift evaluation and uncertainty, as well as engineering reliability, providing different approaches to the development of rubidium fountain clocks and continuously advancing both their performance indicators and practical applications. This article first briefly introduces the operating principle of the rubidium fountain clock, and then summarizes and analyzes the research progress and frequency-stability performance of relevant domestic and international institutions. By elaborating on the distinctive features and technical routes of each institution, the key technologies of the rubidium fountain clock are highlighted. Finally, conclusions and an outlook are given. The rubidium fountain clock should, on the one hand, continuously improve its performance indicators such as frequency stability, and on the other hand, move toward engineering application and promotion through miniaturization and lightweight design.