We discuss the all-important issue of nonlinearity in the physical sciences, with emphasis on its importance in condensed matter physics, where nonlinearity provides a natural mechanism for selective trapping of excitations, such as electrons, at a microscopic scale. This mechanism, term "selftrapping", is investigated in a variety of lattices with different geometries and dimensionalities. It is found that its main features are universal and robust against perturbations of several kind. These findings would have an impact on the future design of smart materials with pre-ordained transport properties, a much sought-after feature nowadays. We also discuss an application of the selftrapping ideas to the problem of power switching in nonlinear optics, an important issue when considering the design of an all-optical computer.