The interface between life and physical sciences provides an abundant habitat for mathematical models. These are often complex to our feeble minds yet ridiculously simplistic in comparison with nature's subtlety. They nevertheless often succeed in extracting important insights into, and sometimes quantitative measures of, nature's ways. The effectiveness of mathematics in the natural sciences was dubbed 'unreasonable' in the title of a famous essay over 50 years ago [1], and is no less so today. The models come in a range of flavours, with elements from deterministic and stochastic mathematics, and from algebra, combinatorics and graph theory among other branches of the mathematical tree. To evaluate the hypotheses about how nature works embodied in such models, their properties-most obviously the outcomes they predict-must be compared with observations. At a minimum, the observations can be at the level of qualitative descriptions of natural systems, but increasingly today models are evaluated on the basis of detailed comparisons with carefully measured data. Moreover, rather than coming at the end of a process of model building and computation, comparisons with data are today usually integrated into every level of the model building process, from choosing the broad features of the model to fine-tuning of parameter values.This issue of Interface Focus is devoted to showcasing some of the many possibilities for innovative approaches to inference in complex models. The emphasis here is not on fully general exposition of methods or rigorous proofs of validity. There is of course some introduction and review of advanced techniques that are more fully developed elsewhere. However, our focus is on application to cutting edge scientific problems. For this showcase, we have selected an unusually broad range of scientific areas, the unifying features being complex models and challenging inference problems tackled by novel techniques.This year marks the centenary of the death of Francis Galton, the great Victorian polymath who has a claim to be regarded as the first applied statistician in the modern sense. He enthusiastically collected and analysed data to illuminate a range of scientific problems from climate to human psychology and morphology, the latter including fingerprints for forensic use. His bequest generated another important centenary this year-the founding of the world's first academic department of statistics at University College London. Academic statistics has flourished in the intervening century, but the number of card-carrying statisticians remains far too small to address the ubiquitous challenges of inference in every branch of science. The toolkit of inference methods available to scientists has constantly been enriched by contributions from other disciplines, including students of insurance, games of chance, psychology, agriculture and genetics. In recent decades, entire schools of inferential thought have evolved in various engineering and informatics disciplines, largely independent of mainstream...