Public health interventions to control the current epidemic of carbapenem-resistant Klebsiella pneumoniae are reliant upon a comprehensive understanding of their emergence and spread over a wide range of geographical scales. We analysed the genome sequences and epidemiological data of >1700 K. pneumoniae, isolated from patients in 244 hospitals in 32 countries, during the European survey of carbapenemase-producing Enterobacteriaceae (EuSCAPE). We demonstrate that carbapenemase acquisition is the main cause of carbapenem resistance and has occurred across diverse phylogenetic backgrounds. However, 477/682 (69.9%) of carbapenemase-positive isolates are concentrated in four clonal lineages, sequence types (ST) 11, 15, 101, 258/512, and their derivatives. Combined analysis of the genetic and geographic distances between isolates with different beta-lactam resistance determinants suggests that the propensity of K. pneumoniae to spread in hospital environments correlates with the degree of resistance and that carbapenemase-positive isolates have the highest transmissibility. Indeed, we found that over half of hospitals contributing carbapenemase-positive isolates likely experienced within-hospital transmission, and inter-hospital spread is far more frequent within, rather than between, countries. Finally, we propose a value of 21 for the number of single nucleotide polymorphisms (SNPs) that optimises discrimination of hospital clusters, and detail the international spread of the successful epidemic lineage, ST258/512.