Large systems of particles interacting via the laws of classical mechanics are widely used models in material science and statistical mechanics. A major topic in this context is the relation between the dynamics of these large, but microscopic systems and a behavior that is in some sense macroscopic. This question was first posed in statistical physics more than 100 years ago and still is one of the most challenging fields of multi-scale analysis. A second category, that is far from thermodynamic fluctuations, is the reversive evolution of initial conditions that are well-defined on the microscopic level. Prototypical problems again concern the passage from discrete lattice dynamics to continuum models describing the effective dynamics on much larger spacial and/or temporal scales. Here, a special case is the long-time behavior of microscopic initial data. In this context, emergence and dynamics of coherent structures, e.g. solitary waves, are of particular interest. The authors cover aspects of both categories by studying a class of systems of infinitly many particles including the celebrated Fermi-Pasta-Ulam chain and the discrete Klein-Gordon equation. The results include sharp decay estimates for the linearized systems, the dispersive stability of small amplitude solutions and numerical studies of the evolution of wave fronts in the nonlinear case, and a possible approach to compute free energies.