We present a model, that while simple to define, shows rich behavior that accounts for a variety of physical phenomena, such as localization of phonons, anomalous diffusion, and slow relaxations in glassy systems. The 'crux of the matter' lies in the fact that the rate of many process es in nature is exponential in the relevant distance: quantum tunneling is a common example. If we think about a particle diffusing in a random environment, its dynamics will be described by a matrix A where the i,j'th element is exponential in the distance between points i and j. This leads us naturally to the model of exponential random matrices, which is a different ensemble of random matrices, with interesting properties. We solve the model exactly in the low density, and discuss the implications on the various physical problems.