The complex multi-criteria optimisation problems arising in Kidney Exchange Programmes have received considerable attention both in practice and in the scientific literature. Whereas theoretical advancements are well reviewed and synthesised, this is not the case for practice. We present a synthesis of models and methods applied in present European Kidney Exchange Programmes, which is based on detailed descriptions we created for this purpose. Most descriptions address national programmes, yet we also present findings on emerging cross-national programmes. The synthesis provides a systematic and detailed description of the models and methods the programmes use, revealing important commonalities as well as considerable variation among them. Rather than distilling a single best practice from these results, we find that the variation in models and methods arises because of variation in country characteristics, policies, and ethics. The synthesised state of the art may benefit future national and cross-national initiatives and direct future theoretical contributions within and across the boundaries of the Operations Research discipline.