Levulinic acid and its esters are close to being extensively produced through consolidated industrial processes, thus playing a central role in biobased industries producing commodities within the principles of the circular economy. One of the main pathways of levulinic acid and ester valorization is their transformation with hydrogen to obtain γ-valerolactone, valeric esters, 1,4-pentanediol and 2-methyl tetrahydrofuran. These reactions are catalyzed by noble and non-noble metal-based heterogeneous catalysts. The use of an abundant and non-toxic element, such as copper, is advantageous with respect to expensive or harmful metals, such as Rh, Ru, Pt or Ni. In this critical review, we wish to give a deeper insight into research advancements in the last ten years regarding the processing of levulinic acid and its esters with hydrogen using heterogeneous copper catalysts.