In this talk, we'll touch on different aspects of discrete systems, as they arise in a variety of recent experiments pertinent to physical applications. We will start from the “traditional” spatial discreteness and how it affects elastic travelling waves in so-called acoustic or granular crystals. We will continue with a form of discreteness (or, more accurately, periodicity) in the evolution variable for pulses passing through a so-called layered optical medium and we will end with an example where the discrete number of components plays a crucial role in the dynamics of two coupled Bose-Einstein condensates in atomic physics. In each example, we will present the relevant dynamical models and we will attempt to mesh the mathematical analysis and numerical computations with experimental results. Finally, we will present some current and near future mathematical, computational and experimental questions of interest along these directions.