Mathematics does not exist in isolation but is linked inextricably to the physical world. At the recent International Congress of Industrial and Applied Mathematics, leading mathematicians from around the globe gathered for a symposium on the 'Mathematics of Real World Problems', which focused on furthering the establishment and dissemination of those links. Presented in four parts, this volume comprises chapters by those invited to this symposium. Part I examines Mathematics for Technology, looking at mathematics as a technology, offering a wide-ranging definition of Industrial Mathematics, and exploring the mathematics of type-II superconductors. Part II provides lucid discussions on theoretical and applied aspects of wavelets, while Part III presents classical and fractal methods for physical problems, including a fractal approach to porous media textures and using MATLAB to model chaos in the motion of a satellite. The final section looks at recent trends in variational methods, exploring areas such as elliptic inverse problems, sweeping processes, and the BBKY hierarchy of quantum kinetic equations. By virtue of its abstraction, mathematics allows the transfer of ideas between fields of applications. 'Mathematical Models and Methods for Real World Systems' clearly demonstrates this and promotes the kind of cross-thinking that nurtures creativity and leads to further innovation.