This overview provides insights into organic and metal-organic polymer (OMOP) catalysts aimed at processes carried out in the liquid phase. Various types of polymers are discussed, including vinyl (various functional poly(styrene-co-divinylbenzene) and perfluorinated functionalized hydrocarbons, e.g., Nafion), condensation (polyesters, -amides, -anilines, -imides), and additional (polyurethanes, and polyureas, polybenzimidazoles, polyporphyrins), prepared from organometal monomers. Covalent organic frameworks (COFs), metal-organic frameworks (MOFs), and their composites represent a significant class of OMOP catalysts. Following this, the preparation, characterization, and application of dispersed metal catalysts are discussed. Key catalytic processes such as alkylation-used in large-scale applications like the production of alkyl--butyl ether and bisphenol A-as well as reduction, oxidation, and other reactions, are highlighted. The versatile properties of COFs and MOFs, including well-defined nanometer-scale pores, large surface areas, and excellent chemisorption capabilities, make them highly promising for chemical, electrochemical, and photocatalytic applications. Particular emphasis is placed on their potential for CO treatment. However, a notable drawback of COF- and MOF-based catalysts is their relatively low stability in both alkaline and acidic environments, as well as their high cost. A special part is devoted to deactivation and the disposal of the used/deactivated catalysts, emphasizing the importance of separating heavy metals from catalysts. The conclusion provides guidance on selecting and developing OMOP-based catalysts.