Currently, a serious threat for living organisms and human life in particular, is water contamination with persistent organic and inorganic pollutants. To date, several techniques have been adopted to remove/treat organics and toxic contaminants. Adsorption is one of the most effective and economical methods for this purpose. Generally, porous materials are considered as appropriate adsorbents for water purification. Conventional adsorbents such as activated carbons have a limited possibility of surface modification (texture and functionality), and their adsorption capacity is difficult to control. Therefore, despite the significant progress achieved in the development of the systems for water remediation, there is still a need for novel adsorptive materials with tunable functional characteristics. This review addresses the new trends in the development of new adsorbent materials. Herein, modern carbon-based materials, such as graphene, oxidized carbon, carbon nanotubes, biomass-derived carbonaceous matrices—biochars as well as their composites with metal-organic frameworks (MOFs) and MOF-derived highly-ordered carbons are considered as advanced adsorbents for removal of hazardous organics from drinking water, process water, and leachate. The review is focused on the preparation and modification of these next-generation carbon-based adsorbents and analysis of their adsorption performance including possible adsorption mechanisms. Simultaneously, some weak points of modern carbon-based adsorbents are analyzed as well as the routes to conquer them. For instance, for removal of large quantities of pollutants, the combination of adsorption and other methods, like sedimentation may be recommended. A number of efficient strategies for further enhancing the adsorption performance of the carbon-based adsorbents, in particular, integrating approaches and further rational functionalization, including composing these adsorbents (of two or even three types) can be recommended. The cost reduction and efficient regeneration must also be in the focus of future research endeavors. The targeted optimization of the discussed carbon-based adsorbents associated with detailed studies of the adsorption process, especially, for multicomponent adsorbate solution, will pave a bright avenue for efficient water remediation.