The Theory-Software Translation Workshop, held in New Orleans in February 2019, explored in depth the process of both instantiating theory in software - for example, implementing a mathematical model in code as part of a simulation - and using the outputs of software - such as the behavior of a simulation - to advance knowledge. As computation within research is now ubiquitous, the workshop provided a timely opportunity to reflect on the particular challenges of research software engineering - the process of developing and maintaining software for scientific discovery. In addition to the general challenges common to all software development projects, research software additionally must represent, manipulate, and provide data for complex theoretical constructs. Ensuring this process is robust is essential to maintaining the integrity of the science resulting from it, and the workshop highlighted a number of areas where the current approach to research software engineering would benefit from an evidence base that could be used to inform best practice. The workshop brought together expert research software engineers and academics to discuss the challenges of Theory-Software Translation over a two-day period. This report provides an overview of the workshop activities, and a synthesises of the discussion that was recorded. The body of the report presents a thematic analysis of the challenges of Theory-Software Translation as identified by workshop participants, summarises these into a set of research areas, and provides recommendations for the future direction of this work.