When metals in supported catalysts are atomically dispersed, they are usually cationic and bonded chemically to supports. Investigations of noble metals in this class are growing rapidly, leading to discoveries of catalysts with new properties. Characterization of these materials is challenging because the metal atoms reside on surfaces that are typically nonuniform in composition and structure. We posit that understanding of structures and catalytic properties of these materials is emerging most strongly from investigations of structurally uniform catalysts (metal atoms dispersed on crystalline supports) which can be characterized incisively with atomic-resolution electron microscopy, X-ray absorption spectroscopy, and infrared spectroscopy, bolstered by density functional theory. We assess the literature of such catalysts supported on zeotype materials, metal-organic frameworks, and covalent organic frameworks. Assessing characterization, reactivity, and catalytic performance of catalysts for oxidation, hydrogenation, the water-gas shift reaction, and others, we consider metal-support interactions and ligand effects for various metal-support combinations, evaluating the degree of structural uniformity of exemplary catalysts and summarizing structure-reactivity and structure-catalytic property relationships.