Lytic polysaccharide monooxygenases (LPMOs) catalyze the oxidative cleavage of glycosidic bonds and represent a promising resource for development of industrial enzyme cocktails for biomass processing. LPMOs show high sequence and modular diversity and are known, so far, to cleave insoluble substrates such as cellulose, chitin and starch, as well as hemicelluloses such as beta-glucan, xyloglucan and xylan. All LPMOs share a catalytic histidine brace motif to bind copper, but differ strongly when it comes to the nature and arrangement of residues on the substrate-binding surface. In recent years, the number of available LPMO structures has increased rapidly, including the first structure of an enzyme–substrate complex. The insights gained from these structures is reviewed below.