Vertebrates at high altitude are subjected to hypoxic conditions that challenge aerobic metabolism. O(2) transport from the respiratory surfaces to tissues requires matching between the O(2) loading and unloading tensions and the O(2)-affinity of blood, which is an integrated function of hemoglobin's intrinsic O(2)-affinity and its allosteric interaction with cellular effectors (organic phosphates, protons and chloride). Whereas short-term altitudinal adaptations predominantly involve adjustments in allosteric interactions, long-term, genetically-coded adaptations typically involve changes in the structure of the haemoglobin molecules. The latter commonly comprise substitutions of amino acid residues at the effector binding sites, the heme-protein contacts, or at intersubunit contacts that stabilize either the low-affinity ('Tense') or the high-affinity ('Relaxed') structures of the molecules. Molecular heterogeneity (multiple isoHbs with differentiated oxygenation properties) can further broaden the range of physico-chemical conditions where Hb functions under altitudinal hypoxia. This treatise reviews the molecular and cellular mechanisms that adapt haemoglobin-oxygen affinities in mammals, birds and ectothermic vertebrates at high altitude.