Due to inherent structural defects, common nanocatalysts always display limited catalytic activity and selectivity, making it practically difficult for them to replace natural enzymes in a broad scope of biologically important applications. By decreasing the size of the nanocatalysts, their catalytic activity and selectivity will be substantially improved. Guided by this concept, the advances of nanocatalysts now enter an era of atomic-level precise control. Single-atom catalysts (denoted as SACs), characterized by atomically dispersed active sites, strikingly show utmost atomic utilization, precisely located metal centers, unique metal-support interactions and identical coordination environments. Such advantages of SACs drastically boost the specific activity per metal atom, and thus provide great potential for achieving superior catalytic activity and selectivity to functionally mimic or even outperform natural enzymes of interest. Although the size of the catalysts does matter, it is not clear whether the guideline of "the smaller, the better" is still correct for developing catalysts at the single-atom scale. Thus, it is clearly a new, urgent issue to address before further extending SACs into biomedical applications, representing an important branch of nanomedicine. This review begins by providing an overview of recent advances of synthesis strategies of SACs, which serve as a basis for the discussion of emerging achievements in improving the enzyme-like catalytic properties at an atomic level. Then, we carefully compare the structures and functions of catalysts at various scales from nanoparticles, nanoclusters, and few-atom clusters to single atoms. Contrary to conventional wisdom, SACs are not the most catalytically active catalysts in specific reactions, especially those requiring multi-site auxiliary activities. After that, we highlight the unique roles of SACs toward biomedical applications. To appreciate these advances, the challenges and prospects in rapidly growing studies of SACs-related catalytic nanomedicine are also discussed in this review.