The Fontan surgery for single ventricle heart defects is a typical example of a clinical intervention in which patient-specific computational modeling can improve patient outcome: with the functional heterogeneity of the presenting patients, which precludes generic solutions, and the clear influence of the surgically-created Fontan connection on hemodynamics, it is acknowledged that individualized computational optimization of the post-operative hemodynamics can be of clinical value. A large body of literature has thus emerged seeking to provide clinically relevant answers and innovative solutions, with an increasing emphasis on patient-specific approaches. In this review we discuss the benefits and challenges of patient-specific simulations for the Fontan surgery, reviewing state of the art solutions and avenues for future development. We first discuss the clinical impact of patient-specific simulations, notably how they have contributed to our understanding of the link between Fontan hemodynamics and patient outcome. This is followed by a survey of methodologies for capturing patient-specific hemodynamics, with an emphasis on the challenges of defining patient-specific boundary conditions and their extension for prediction of post-operative outcome. We conclude with insights into potential future directions, noting that one of the most pressing issues might be the validation of the predictive capabilities of the developed framework.