Weather phenomena can result in severe impacts on railway infrastructure. In future, projected changes to the frequency and/or intensity of extreme weather events could change weather–infrastructure risk profiles. Infrastructure owners and operators need to manage current weather impacts and put in place adequate plans to anticipate and adapt to changes in future weather risks, or mitigate the impacts arising from those risks. The assessment of the risk posed to railway infrastructure from current and future weather is dependent on a good understanding of the constituent components of risk: hazard, vulnerability, and exposure. A good understanding of the baseline and projected future risk is needed in order to understand the potential benefits of various climate change adaptation actions. Traditional risk assessment methods need some modification in order to be applied to climate change timescales, for which decisions need to be made under deep uncertainty. This review paper highlights some key challenges for assessing the risk, including: managing uncertainties; understanding weather-impact relationships and how they could change with climate change; assessing the costs of current and future weather impacts and the potential cost versus benefit of adaptation; and understanding practices and tools for adapting railway infrastructure. The literature reveals examples of progress and good practice in all these areas, providing scope for effective knowledge-sharing—across the railway infrastructure and other sectors—in support of infrastructure resilience and adaptation.