Furandicarboxylic acid (FDCA) has been considered as a good precursor, instead of petroleum-derived terephthalate acid (TPA), for producing green polymers such as polyethylene 2,5-furandicarboxylate (PEF). The production of FDCA from biomass or its derived sugars or platform chemicals generally involves chemical, biological and electrochemical methods, while the chemical-catalytic way seems to be the most promising in terms of the yield, reaction rate and product purity. The oxidative production of FDCA from bio-based 5-hydroxymethylfurfural (HMF) has attracted the most attention; it can be carried out by electrochemical, catalytic and non-catalytic processes. In the present work, we have comprehensively reviewed the current progress on the production of FDCA from HMF, primarily focusing on the chemical-catalytic approaches. The most frequently used catalysts for the chemical-catalytic methods are oxides of noble metals but their high cost, poor availability and recycling are the major hindrances to their commercial acceptance. Transition metal oxides are good alternatives but they suffer from relatively low FDCA yield. Electrochemical oxidation of HMF can be a good alternative route for FDCA production with simultaneous H₂ production but the yield and product recovery have to be further improved. Biocatalytic processes can produce FDCA with comparative yields under mild conditions but they can only be operated at low concentrations of HMF with much lower productivity. It is recommended that future works should be focused on, but not limited to, the comprehensive evaluation of different routes in terms of catalyst development and characterizations, process parameters, product yield and purity, as well as economic feasibility. The kinetics and reaction mechanisms of the process also need to be more deeply investigated to guide further process intensification.