Adsorption and desorption play major roles in separations, purification of water, waste streams, liquid fuels, catalysis, biomedicine and chromatography. Mesoporous metal-organic frameworks (MOFs) with pore sizes 2-50 nm are particularly suitable for adsorption of organic compounds in solution. Tens of thousands of aromatic and heterocyclic compounds are major components of liquid fuels, feedstock for industrial synthesis, solvents, dyestuffs, agricultural chemicals, medicinal drugs, food additives, and so forth. This Review provides a systematization and analysis of studies on adsorption/desorption on mesoporous MOFs in solution and their underlying chemical mechanisms. The (in)stability of mesoporous MOFs in water is critically discussed. Adsorption capacity and selectivity are covered for organic dyes, medicinal drugs, major components of liquid fuels, and miscellaneous industrial chemicals. Ionic interactions, Brønsted acid-base interactions, hydrogen bonding, coordination bonding, π-π interactions, and non-specific interactions are covered amongst adsorption mechanisms. The effects of post-synthetic modifications of mesoporous MOFs on their stability, adsorption capacity, selectivity, and mechanisms of adsorption and desorption are analyzed. To encourage research in this quickly growing field, we identify "niches" for which no application-oriented and/or mechanistic studies were reported. Perspectives and limitations of a wide use of mesoporous MOFs as industrial sorbents are discussed.