The massive burning of a large amount of fossil energy has caused a lot of serious environmental issues (e.g., air pollution and climate change), urging people to efficiently explore and valorize sustainable alternatives. Biomass is being deemed as the only organic carbon-containing renewable resource for the production of net-zero carbon emission fuels and fine chemicals. Regarding this, the selective transformation of high-oxygen biomass feedstocks by catalytic transfer hydrogenation (CTH) is a very promising strategy to realize the carbon cycle. Among them, the important Meerwein-Ponndorf-Verley (MPV) reaction is believed to be capable of replacing the traditional hydrogenation strategy which generally requires high-pressure H2 and precious metals, aiming to upgrade biomass into downstream biochemical products and fuels. Employing bifunctional heterogeneous catalysts with both acidic and basic sites is needed to catalyze the MPV reaction, which is the key point for domino/cascade reaction in one pot that can eliminate the relevant complicated separation/purification step. Zirconium (Zr) and hafnium (Hf), belonging to transition metals, rich in reserves, can demonstrate similar catalytic efficiency for MPV reaction as that of precious metals. This review introduced the application of recyclable heterogeneous non-noble Zr/Hf-containing catalysts with acid-base bifunctionality for CTH reaction using the safe liquid hydrogen donor. The corresponding catalysts were classified into different types including Zr/Hf-containing metal oxides, supported materials, zeolites, metal-organic frameworks, metal-organic hybrids, and their respective pros and cons were compared and discussed comprehensively. Emphasis was placed on evaluating the bifunctionality of catalytic material and the key role of the active site corresponding to the structure of the catalyst in the MPV reaction. Finally, a concise summary and prospect were also provided centering on the development and suggestion of Zr/Hf-containing acid-base bifunctional catalysts for CTH.