The nanoparticle (NP) lies at the very heart of nanotechnology. The wide range of current and potential uses of NPs, which include medicine, manufacturing, environment, and energy, has stimulated an international research effort focusing on NP synthesis and design. Despite extensive research into NPs, many breakthroughs and advances are the result of trial and error, since the mathematical framework which could provide a theoretical understanding often does not exist.This is partly because classical theories of heat transfer and phase change, which are relevant to the design and use of NPs, breakdown on the nanoscale.The focus of the proposed research is, therefore, to develop and analyse novel mathematical models describing nanoscale heat transfer and phase change. Exact, approximate, and numerical solutions will be obtained in order to gain practical insights into NP evolution. To ensure the models and results are realistic and of use to nanoscience researchers, the work will be carried out in collaboration with members of the Inorganic Nanoparticles Group of the Catalan Institute for Nanoscience and Nanotechnology, and the Statistical Physics Group at the Autonomous University of Barcelona. The project, which is truly multi-disciplinary, will bridge the gap between mathematics and nanotechnology.