Furans represent one of the most important classes of intermediates in the conversion of non-edible lignocellulosic biomass into bio-based chemicals and fuels. At present, bio-furan derivatives are generally obtained from cellulose and hemicellulose fractions of biomass via the acid-catalyzed dehydration of their relative C6-C5 sugars and then converted into a wide range of products. Furfural (FUR) and 5-hydroxymethylfurfural (HMF) are surely the most used furan-based feedstocks since their chemical structure allows the preparation of various high-value-added chemicals. Among several well-established catalytic approaches, hydrogenation and oxygenation processes have been efficiently adopted for upgrading furans; however, harsh reaction conditions are generally required. In this review, we aim to discuss the conversion of biomass derived FUR and HMF through unconventional (transfer hydrogenation, photocatalytic and electrocatalytic) catalytic processes promoted by heterogeneous catalytic systems. The reaction conditions adopted, the chemical nature and the physico-chemical properties of the most employed heterogeneous systems in enhancing the catalytic activity and in driving the selectivity to desired products are presented and compared. At the same time, the latest results in the production of FUR and HMF through novel environmental friendly processes starting from lignocellulose as well as from wastes and by-products obtained in the processing of biomass are also overviewed.