The palliation of the cyanotic child with a dominant morphologically left ventricle, discordant ventriculo-arterial connections, and obstruction to the pulmonary outflow tract has continued to evolve and mature. The evolution began in the early days of surgical palliation with the Blalock-Taussig shunt, extended to construction of cavopulmonary shunts, if required, and then to the Fontan procedure and its subsequent modifications. This journey took nearly 30 years to complete. There is increasing clinical data to document the beneficial effects of this approach, with ever-improving outcomes. Some aspects of the history of the cavopulmonary shunt have been previously reviewed in this journal and elsewhere, as have analysis of outcomes for some groups of patients considered for surgical completion of the Fontan circulation. While there has been some ongoing interest in ventricular septation since the early success of Sakakibara et al., this approach has largely been abandoned. Considerably more challenges and debate resonate in the surgical algorithms defined for patients whose hearts are characterized by a dominant left ventricle, discordant ventriculo-arterial connections, and unobstructed flow of blood to the lungs. This latter group will be the focus of this review, as will the aetiology of the myocardial hypertrophy that is particularly frequent in this group of patients, its clinical recognition, indeed its anticipation, and the multiple surgical strategies designed to prevent or treat it. All these manoeuvres are considered to optimise suitability for, and outcome from, creation of the Fontan circulation.