Climate change, socioeconomical pressures, and new policy and legislation are driving a decarbonization process across industries, with a critical shift from a fossil-based economy toward a biomass-based one. This new paradigm implies not only a gradual phasing out of fossil fuels as a source of energy but also a move away from crude oil as a source of platform chemicals, polymers, drugs, solvents and many other critical materials, and consumer goods that are ubiquitous in our everyday life. If we are to achieve the United Nations’ Sustainable Development Goals, crude oil must be substituted by renewable sources, and in this evolution, biorefineries arise as the critical alternative to traditional refineries for producing fuels, chemical building blocks, and materials out of non-edible biomass and biomass waste. State-of-the-art biorefineries already produce cost-competitive chemicals and materials, but other products remain challenging from the economic point of view, or their scaled-up production processes are still not sufficiently developed. In particular, lignin’s depolymerization is a required milestone for the success of integrated biorefineries, and better catalysts and processes must be improved to prepare bio-based aromatic simple molecules. This review summarizes current challenges in biorefinery systems, while it suggests possible directions and goals for sustainable development in the years to come.