Covalent organic frameworks (COFs) are a class of crystalline porous organic polymers with permanent porosity and highly ordered structures. Unlike other polymers, a significant feature of COFs is that they are structurally predesignable, synthetically controllable, and functionally manageable. In principle, the topological design diagram offers geometric guidance for the structural tiling of extended porous polygons, and the polycondensation reactions provide synthetic ways to construct the predesigned primary and high-order structures. Progress over the past decade in the chemistry of these two aspects undoubtedly established the base of the COF field. By virtue of the availability of organic units and the diversity of topologies and linkages, COFs have emerged as a new field of organic materials that offer a powerful molecular platform for complex structural design and tailor-made functional development. Here we target a comprehensive review of the COF field, provide a historic overview of the chemistry of the COF field, survey the advances in the topology design and synthetic reactions, illustrate the structural features and diversities, scrutinize the development and potential of various functions through elucidating structure-function correlations based on interactions with photons, electrons, holes, spins, ions, and molecules, discuss the key fundamental and challenging issues that need to be addressed, and predict the future directions from chemistry, physics, and materials perspectives.