Using nanomaterials to mimic the function of protein enzymes is an interesting idea. Many nanomaterials have a similar size as enzymes and they also possess catalytic activity. Over the past decade, a surge of nanozyme work has emerged, likely due to the advancement in the synthesis and characterization of inorganic nanoparticles. Many typical enzymatic reactions mimicking oxidases, peroxidases, laccases, superoxide dismutases, and catalases have been realized by simple metal oxide and metal nanoparticles. In addition, small inorganic catalysts have been loaded in nanoparticles to create another type of nanozyme. The applications of nanozymes in biosensor design, environmental remediation, and therapeutics have been demonstrated. In this Topical Review, we briefly summarize the current status of the field and then focus our attention on some important problems faced by the field. These topics include developing better nanozymes with higher activity, better substrate selectivity, and engineering enzyme-like active sites. For practical applications, reliable methods for bioconjugation of nanozymes with affinity ligands need to be achieved, but not at the cost of losing the activity of nanozymes. Finally, fundamental mechanistic studies are needed to rationally design nanozymes and to obtain key insights into a few model systems.