Interest in lignocellulosic biomass conversion technologies has increased recently because of their potential to reduce the dependency on non-renewable feedstocks. Residues from a variety of crops are the major source of lignocellulose, which is being produced in increasingly large quantities worldwide. The commercial exploitation of crop residues as feedstocks for biorefineries which could be used to produce a variety of goods such as biofuels, biochemicals, bioplastics, and enzymes is an attractive approach not only for adding value to residues but also for providing renewable products required by the expanding bioeconomy market. Moreover, the implementation of biorefineries in different regions has the potential to add value to the specific crop residues produced in the region. In this review, several aspects of crop residue application in biorefineries are discussed, including the role of crop residues in the bioeconomy and circular economy concepts, the main technical aspects of crop residue conversion in biorefineries, the main crop residues generated in different regions of the world and their availability, the potential value-added bioproducts that can be extracted or produced from each crop residue, and the major advantages and challenges associated with crop residue utilization in biorefineries. Despite their potential, most biomass refining technologies are not sufficiently advanced or financially viable. Several technical obstacles, especially with regard to crop residue collection, handling, and pre-treatment, prevent the implementation of biorefineries on a commercial scale. Further research is needed to resolve these scale-up-related challenges. Increased governmental incentives and bioeconomic strategies are expected to boost the biorefinery market and the cost competitiveness of biorefinery products.