Tsunami damage, fragility, and vulnerability functions are statistical models that provide an estimate of expected damage or losses due to tsunami. They allow for quantification of risk, and so are a vital component of catastrophe models used for human and financial loss estimation, and for land-use and emergency planning. This paper collates and reviews the currently available tsunami fragility functions in order to highlight the current limitations, outline significant advances in this field, make recommendations for model derivation, and propose key areas for further research. Existing functions are first presented, and then key issues are identified in the current literature for each of the model components: building damage data (the response variable of the statistical model), tsunami intensity data (the explanatory variable), and the statistical model that links the two. Finally, recommendations are made regarding areas for future research and current best practices in deriving tsunami fragility functions (see Discussion, Recommendations, and Future Research). The information presented in this paper may be used to assess the quality of current estimations (both based on the quality of the data, and the quality of the models and methods adopted) and to adopt best practice when developing new fragility functions.