The Tasmanian tiger or thylacine (Thylacinus cynocephalus) was the largest carnivorous Australian marsupial to survive into the modern era. Despite last sharing a common ancestor with the eutherian canids ~160 million years ago, their phenotypic resemblance is considered the most striking example of convergent evolution in mammals. The last known thylacine died in captivity in 1936 and many aspects of the evolutionary history of this unique marsupial apex predator remain unknown. Here we have sequenced the genome of a preserved thylacine pouch young specimen to clarify the phylogenetic position of the thylacine within the carnivorous marsupials, reconstruct its historical demography and examine the genetic basis of its convergence with canids. Retroposon insertion patterns placed the thylacine as the basal lineage in Dasyuromorphia and suggest incomplete lineage sorting in early dasyuromorphs. Demographic analysis indicated a long-term decline in genetic diversity starting well before the arrival of humans in Australia. In spite of their extraordinary phenotypic convergence, comparative genomic analyses demonstrated that amino acid homoplasies between the thylacine and canids are largely consistent with neutral evolution. Furthermore, the genes and pathways targeted by positive selection differ markedly between these species. Together, these findings support models of adaptive convergence driven primarily by cis-regulatory evolution.