Furfural and 5-hydroxymethylfurfural stand out as bridges connecting biomass raw materials to the biorefinery industry. Their reductive transformations by hydroconversion are key routes toward a wide variety of chemicals and biofuels, and heterogeneous catalysis plays a central role in these reactions. The catalyst efficiency highly depends on the nature of metals, supports, and additives, on the catalyst preparation procedure, and obviously on reaction conditions to which catalyst and reactants are exposed: solvent, pressure, and temperature. The present review focuses on the roles played by the catalyst at the molecular level in the hydroconversion of furfural and 5-hydroxymethylfurfural in the gas or liquid phases, including catalytic hydrogen transfer routes and electro/photoreduction, into oxygenates or hydrocarbons (e.g., furfuryl alcohol, 2,5-bis(hydroxymethyl)furan, cyclopentanone, 1,5-pentanediol, 2-methylfuran, 2,5-dimethylfuran, furan, furfuryl ethers, etc.). The mechanism of adsorption of the reactant and the mechanism of the reaction of hydroconversion are correlated to the specificities of each active metal, both noble (Pt, Pd, Ru, Au, Rh, and Ir) and non-noble (Ni, Cu, Co, Mo, and Fe), with an emphasis on the role of the support and of additives on catalytic performances (conversion, yield, and stability). The reusability of catalytic systems (deactivation mechanism, protection, and regeneration methods) is also discussed.