After isolation of radioactive waste in deep geological formations, radionuclides can enter the biosphere via slow migration through engineered barriers and host rocks. The amount of radionuclides that migrate into the biosphere depends on the distance from a repository, dominant transport mechanism (diffusion vs. advection), and interaction of dissolved radionuclides with minerals present in the host rock and engineered barrier systems. Within the framework of the European Union’s Horizon 2020 EURAD project (https://www.ejp-eurad.eu/), a series of state-of-the-art reports, which form the basis of a series of papers, have been drafted. This state-of-the-art paper aims to provide non-specialists with a comprehensive overview of the current understanding of the processes contributing to the radionuclide retention and migration in clay and crystalline host rocks, in a European context. For each process, a brief theoretical background is provided, together with current methodologies used to study these processes as well as references for key data. Owing to innovative research on retention and migration and the extensive knowledge obtained over decades (in the European context), process understanding and insights are continuously improving, prompting the adaptation and refinement of conceptual descriptions regarding safety assessments. Nevertheless, there remains important research questions to be investigated in the future.