Skewed masonry arch bridges form an important part of the rail and road infrastructure networks in the UK, as well as other European countries. However, the precise flow of forces in these structures is not well understood, which can pose a problem when it comes to assessment and maintenance. The skewed masonry arch railway bridge considered in this study has suffered significant historic damage, which led to a pronounced response under live loads and prompted extensive repair work. Subsequently, a network of fibre-optic Fibre Bragg Grating sensors was installed on the arch barrel of the repaired bridge. In this paper, the monitoring method is outlined, and the detailed measurement of the dynamic response under train loading is presented. Results quantify both the principal directions of strain and the strain magnitude in a skewed arch barrel during the passage of a train. Thus, the monitoring data provide rare insight into the structural response of skewed masonry arch bridges.