Abstract: Despite thousands of years of practical experience in metal working, the plastic deformation properties of crystalline materials, such as metals, are still not satisfactorily understood and the subject of intense current research activity. Since the work of V. Volterra and G.I. Taylor, it has been known that the predominant mechanism allowing crystalline solids to deform is the movement of defects in the crystal lattice in the form of so-called "dislocation lines". When a dislocation line moves, it causes a net displacement of crystal planes and thus a (very small) deformation of the macroscopic specimen. Yet, even after a century of research, we still do not have a comprehensive, reliable, and rigorous theory that describes the plastic deformation of macroscopic crystalline solids from its micro-mechanical principles. This goal is of such practical, as well as theoretical, importance that it is sometimes referred to as the “Holy Grail” of Solid Mechanics. This talk will survey several aspects of my work on microscopic and macroscopic elasto-plasticity and also present some surprising connections to other fields of mathematics, both applied and theoretical.