In recent years, advanced oxidation process (AOPs) based on sulfate radical (SO4●−) and singlet oxygen (1O2) has attracted a lot of attention because of its characteristics of rapid reaction, efficient treatment, safety and stability, and easy operation. SO4●− and 1O2 mainly comes from the activation reaction of peroxymonosulfate (PMS) or persulfate (PS), which represent the oxidation reactions involving radicals and non-radicals, respectively. The degradation effects of target pollutants will be different due to the type of oxidant, reaction system, activation methods, operating conditions, and other factors. In this paper, according to the characteristics of PMS and PS, the activation methods and mechanisms in these oxidation processes, respectively dominated by SO4●− and 1O2, are systematically introduced. The research progress of PMS and PS activation for the degradation of organic pollutants in recent years is reviewed, and the existing problems and future research directions are pointed out. It is expected to provide ideas for further research and practical application of advanced oxidation processes dominated by SO4●− and 1O2.