This year marks the fiftieth anniversary of the publication of the first study of the Fermi-Pasta-Ulam (FPU) problem. The results of that study, which were initially characterized by Fermi as a "little discovery, " in fact heralded the beginning of computational and (modern) nonlinear physics. The work marked the first systematic study of a nonlinear system by digital computers ("experimental mathematics") and led directly to the discovery of "solitons," as well as to deep insights into deterministic chaos and statistical mechanics. In this colloquium, I recall the original FPU problem and show how a multiple-scale analysis in the continuum limit leads to the prediction of the stable, propagating nonlinear excitations-the solitons. I then describe how a similar multiple-scale analysis combined with computational studies led to some seemingly paradoxical results about the existence and stability of "breathers"-spatially localized, time-periodic solutions-in continuum nonlinear systems. The resolution of these paradoxes was the discovery, in the early 1990s, of stable breathers in discrete nonlinear systems. These "discrete breathers"-now more commonly known as Intrinsic Localized Modes (ILMs)-remained only an appealing theoretical possibility for nearly a decade. I review the basic mechanism that allows the existence of ILMs and discuss some of their essential features, including their occurrence in discrete systems in any number of spatial dimensions. In the past few years, this theoretical possibility has become experimental reality. I review recent experiments that have observed ILMs in physical systems as distinct as charge-transfer solids, Josephson junction arrays, photonic structures, and micromechanical oscillator arrays. In closing, I indicate possible future directions for research on ILMs and potential applications of these novel nonlinear excitations.

ANL Physics Division Colloquium Schedule