The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in non-metallic solids and develop "experimental" techniques for studying nonlinear problems, continues to yield a walth of results in the theory and application of nonlinear waves and (many degree-of-freedom) Hamiltonian dynamics. For example, it led to the mathematical "discovery" of solitons and was the model problem employed in the first application of Chirikov's overlap criterion (to estimate critical values of a perturbation required for a system to transition from local to global chaos). This informal talk, which should take 2 lectures, concerns two interweaving stories and will consist primarily of introductory material on solitary waves and the FPU problem. Time-permitting, I will also discuss related research by my collaborators on nonlinear lattice dynamics of Bose-Einstein condensates, which they and I reviewed in a recent article written for _Chaos_'s special issue on the FPU problem.