It is clear that the sizes of chemical, 'drug-like', and materials spaces are enormous. If scientists working in established therapeutic, and newly established regenerative medicine fields are to discover better molecules or materials, they must find better ways of probing these enormous spaces. There are essentially five ways that this can be achieved: combinatorial and high throughput synthesis and screening approaches; fragment-based methods; de novo molecular design, design of experiments, diversity libraries; supramolecular approaches; evolutionary approaches. These methods either synthesise materials and screen them more quickly, or constrain chemical spaces using biology or other types of 'fitness functions'. High throughput experimental approaches cannot explore more than a minute part of chemical space. We are nevertheless entering into an era that is data dominated. High throughput experiments, robotics, automated crystallographic beam lines, combinatorial and flow synthesis, high content screening, and the 'omics' technologies are providing a flood of data, and efficient methods for extracting meaning from it are essential. This paper describes how new developments in mathematics have provided excellent, robust computational modelling tools for exploring large chemical spaces, for extracting meaning from large datasets, for designing new bioactive agents and materials, and for making truly predictive, quantitative models of the properties of molecules and materials for use in therapeutic and regenerative medicine. We describe these broadly applicable modelling tools and provide examples of their application to serum free stem cell culture, pathogen resistant polymers for implantable devices, new markers and biological mechanisms derived from mathematical analyses of gene array data, and pharmacokinetically important physicochemical properties of small molecules. We also discuss biologically conserved peptide motifs as a design framework for small molecule drugs and give examples of the application of this concept to drug design.