The regression discontinuity design is an elegant and effective way to draw causal conclusions, with an abundance of successful applications in economics, education, political science, social and behavioural sciences and many other areas. Yet it has recently come into focus that the regression discontinuity approach is very underutilized in medical research. This may be due in part because there are still gaps in available methodology for common medical data structures. In this project, we aim to fill these gaps by tailoring recently developed mathematical insights and tools from Bayesian statistics to sharp and fuzzy regression discontinuity designs. In the first work package, a novel Bayesian method to study survival data in a regression discontinuity design is developed and placed on a firm theoretical foundation. A multiscale approach will be adopted, and the priors for the hazard will be of piecewise constant form to allow for flexible inclusion of covariates while preserving reliable uncertainty quantification. Second, we aim to combine evidence from multiple hospitals, with the cut-off varying from hospital to hospital. The local estimates from each center will be combined in a nonparametric Bayesian regression approach using the BART prior, which has previously found great success in other statistical problems. Theoretical results will be derived to justify the accompanying uncertainty quantification. The end result will be a flexible method to combine evidence, which can be used for continuous and survival data. Third, we design and rigorously evaluate (through posterior contraction and Bernstein-Von Mises theorems) a prior distribution for the situation where it is unclear what the cut-point is, or whether a cut-point was used at all. The Bayesian approach is highly promising here since it yields automatic uncertainty quantification, and in case multiple data sets are available, it allows for borrowing of strength between the data sets in a natural way. Finally, data from the Dutch Arthroplasty Register is analysed in close collaboration with orthopaedic surgeons, to study which type of fixation is optimal in terms of hip implant survival. This is the application that inspired the three aforementioned work packages. Interwoven with all work packages is the development of an R package and a workshop, so that the novel methods will become easily accessible to researchers working in a variety of research fields. The overall result of all work packages combined will be that medical as well as other researchers can apply the regression discontinuity design in more settings, opening up more observational data sets for causal inference.