Spatial transformations: from fundamentals to applications.

This paper forms the introduction to this themed issue of Philosophical Transactions of the Royal Society A on 'Spatial transformations', arising from the Royal Society Scientific Discussion Meeting held in January 2015. The paper begins with a review of the concepts and history of spatial transformations, followed by a discussion of the contributions from the papers in this themed issue. A summary of the advantages and current limitations of spatial transformations concludes the paper, with the key challenges identified at the Scientific Discussion Meeting also given.