Society faces major challenges from viral diseases. The recent Zika and Ebola outbreaks are only two examples of the devastating impact of viral illnesses on human health, and viral pathogens infecting agriculturally important livestock and plants simultaneously reduce food production and inflict great annual financial losses worldwide. Viruses, however, also have positive impacts on health and ecology. They balance and stabilise our gut microbiome, preventing serious illnesses such as certain autoimmune diseases, and influence our climate owing to their roles in carbon cycling in the oceans. It is therefore paramount to better understand virus structure and function across the entire virosphere in order to control, and even take advantage of, viruses in medicine and biotechnology. I have demonstrated previously that mathematical approaches developed in tandem with experimentalists are drivers of discovery of functionally crucial structural viral features, revealing their novel functional roles in viral life cycles, and enabling their exploitation in therapy and biotechnology. Previously developed mathematical approaches were geared towards a specific major sub-group of the virosphere. In this research programme, I will both broaden and deepen the development of novel mathematical techniques. Working in close collaboration with leading experimental groups, at a larger scale, I will identify functionally important geometric viral features in a number of major groups of viruses. This will include: geometric strand assortment in multipartite viruses, such as the major agricultural pathogen Bluetongue virus; the assembly of retroviruses like HIV, with applications to the construction of virus-like particles from viral components as vectors for gene editing and therapy; and the structure and evolution of viruses important for the gut microbiome and marine ecology. By linking structural features with their functions, I will address open problems regarding drivers of evolution in one of the simplest yet most important groups of biological entities. This approach will unmask evolutionarily conserved functional features that can be used as novel targets in anti-viral therapy, for the development of novel safer vaccines or repurposed in bionanotechnology.