Furfural and hydroxymethylfurfural are promising platform molecules for manufacturing chemicals and fuel components. These furanic compounds are the product of the acid-catalyzed dehydration of sugars (e.g., xylose and glucose), components obtained from lignocellulosic biomass. Manufacturing furans employs the use of mineral acid catalysts (e.g., H2SO4) in an aqueous medium. This approach limits the selectivity towards furans to approx. 45 mol%, mainly by the formation of solid by-products (humins). The use of aqueous-organic biphasic conditions raises the selectivity to approx. 60-70 mol%. However, even higher selectivities (>80 mol%) can be achieved by switching to organic solvent systems. Specifically, aprotic polar organic solvents (e.g., DMSO) can improve both the conversion and the selectivity from sugars to platform molecules. The presence of aprotic polar organic solvents has an influence on the solvent shell of the sugar and on the activity of the catalyst. Studying these two effects is crucially important to understand improvements of the selectivity. The aim of this review is to explore the use of polar organic solvents in the process of the manufacture of furans, while addressing the challenges of its industrial application, particularly in solvent recovery and recycling.