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Environmental regulatory agencies have implemented stringent restrictions on the permissible levels of sulfur compounds in fuel to reduce harmful emissions and improve air quality. Problematically, traditional desulfurization methods have shown low effectiveness in the removal of refractory sulfur compounds, e.g., thiophene (TS), dibenzothiophene (DBT), and 4-methyldibenzothiophene (MDBT). In this work, molecular dynamics (MD) simulations and free energy perturbation (FEP) have been applied to investigate the use of ionic liquids (ILs) and deep eutectic solvents (DESs) as efficient TS/DBT/MDBT extractants. For the IL simulations, the selected cation was 1-butyl-3-methylimidazolium [BMIM], and the anions included chloride [Cl], thiocyanate [SCN], tetrafluoroborate [BF4], hexafluorophosphate [PF6], and bis(trifluoromethylsulfonyl)amide [NTf2]. The DESs were composed of choline chloride with ethylene glycol (CCEtg) or with glycerol (CCGly). Calculation of excess chemical potentials predicted the ILs to be more promising extractants with energies lower by 1-3 kcal/mol compared to DESs. Increasing IL anion size was positively correlated to enhanced solvation of S-compounds, which was influenced by energetically dominant solute-anion interactions and favorable solute-[BMIM] pi-pi stacking. For the DESs, the solvent components offered a range of synergistic, yet comparatively weaker electrostatic interactions that included hydrogen bonding and cation-pi interactions. An in-depth analysis of the structure of IL and DES systems is presented, along with a discussion of the critical factors behind experimental trends of S-compound extraction efficiency.