The first human liver transplant operation was performed by Thomas Starzl in 1963. The next two decades were marked by difficulties with donor organ quality, recipient selection, operative and perioperative management, immunosuppression and infectious complications. Advances in each of these areas transformed liver transplantation from an experimental procedure to a standard treatment for end-stage liver disease and certain cancers. From the handful of pioneering programmes, liver transplantation has expanded to hundreds of programmes in >80 countries. 1-year patient survival rates have exceeded 80% and outcomes continue to improve. This success has created obstacles. Ongoing challenges of liver transplantation include those concerning donor organ shortages, recipients with more advanced disease at transplant, growing need for retransplantation, toxicities and adverse effects associated with long-term immunosuppression, obesity and NASH epidemics, HCV recurrence and the still inscrutable biology of hepatocellular carcinoma. This Perspectives summarizes this transformation over time and details some of the challenges ahead.