Mathematical models are widely used to inform high-impact decisions for humanity, such as responding to climate change, managing the economy, predicting and controlling renewable energy systems, and dealing with the COVID- 19 pandemic. Improvements in computing power and accessibility and new developments in machine learning have made sophisticated modelling machinery widely available even to people who are not well-acquainted with theoretical fundamentals of modelling and simulation. This presents risks: both risks of poor-quality modelling informing poor decision-making with high real-world impact, and also risks relating to the erosion of public trust in scientific information. These risks can be reduced by improving the conceptual and mathematical foundation on which socially-relevant modelling endeavours are based. While this is a critical justification in itself, there is also a huge opportunity to develop new mathematical tools for the next generation of models, to share recent developments and to bring together and nurture pockets of good practice from across modelling disciplines and areas of application, united by a common mathematical approach.