Lytic polysaccharide monooxygenases (LPMOs) are a new class of microbial copper enzymes involved in the degradation of recalcitrant polysaccharides. They have only been discovered and characterized in the last 5–10 years and have stimulated strong interest both in biotechnology and in bioinorganic chemistry. In biotechnology, the hope is that these enzymes will finally help to make enzymatic biomass conversion, especially of lignocellulosic plant waste, economically attractive. Here, the role of LPMOs is likely to be in attacking bonds that are not accessible to other enzymes. LPMOs have attracted enormous interest since their discovery. The emphasis in this review is on the past and present contribution of crystallographic studies as a guide to functional understanding, with a final look towards the future.