Artificial enzymes are a class of catalysts being potentially viable alternatives to natural enzymes. Enzyme mimics can have the important advantages including tunable structures and catalytic efficiencies, stability in experimental conditions, lower cost, and simple synthetic routes to their preparation. Cyclodextrins, metal complexes, porphyrins, polymers, dendrimers, membranes, nanoparticles, organic and metalorganic compounds and biomolecules can serve as the basis for an artificial catalyst. On the path of mimicry, seven stages have been formulated. Typical examples of enzyme mimetic catalysts, performance, the reactions they catalyze, and the main source of their catalytic efficiency were considered. Catalytically active nanomaterials (nanozymes) also show several advantages over natural enzymes. A wide range of materials are used as nanoenzymes, such as nanoparticles of Fe3O4, Prussian blue, Mn3O4, CeO2, WO2, NiCo2O4@MnO2, graphene oxide, graphene-hemin nanocomposites, carbon nanotubes, carbon nanodots, mesoporous silica-encapsulated gold nanoparticles, gold nanoclusters, Pep–Au-NP = peptide–gold nanoparticle conjugate gold nanoparticles, ferrous ferrocyanide, nonheme iron, vanadium, and copper complexes bearing hemicryptophane. Metal atom (M = K, Ti, Fe, Co, Ni, Cu, Rh) doping on a Mo6S8 cluster and mononuclear rhodium species anchored on a zeolite or titanium dioxide illustrate single-atom catalysis. Nanozymes can exhibit activity of oxidase, catalase, superoxide dismutase, DNAse, alkaline phosphatase phospholipase, topoisomerase and can display antioxidant and biofilm activity. Molecular imprinting is a technique to create template (analogs of substrates, transition states, or products) shaped cavities in polymer matrices with predetermined selectivity and high affinity in a specific function. Schemes of the principle of molecular imprinting recognition, a molecularly imprinted catalyst scheme for selective catalysis, and, as an illustration, imprinted catalytic system based on methacrylic layer polymerization were presented. Despite the obvious advantages of artificial enzymes, it is still not possible to achieve the activity of natural catalysts.