In the past few decades, significant advances in computational modeling of materials on multiple length and time scales have brought about the realization that the critical questions are often encountered on the mesoscale: the length and time scales much larger than atomic, yet much smaller than macroscopic observables. This is the focus area of the course. We select four broad areas representative of the field in the sense that: (a) Significant progress has been made in the past two decades, (b) Some outstanding problems remain as a challenge for future researchers, and, (c) The past efforts have been multidisciplinary, and future efforts are expected to remain such, involving researchers in engineering disciplines, physics and mathematics. - Dislocation plasticity is an extremely challenging problem, owing to: geometrical and statistical complexity of dislocation assemblies, long-range interactions of dislocations, and, still poorly understood dislocation – interface interactions. Several levels of mathematical analysis have emerged over the years involving different level of approximation and computational costs. - Structure and motion of interfaces in solids and fluids, and their role in the deformation of polycrystalline assemblies remain active area of research. For problems with mobile interfaces and topological discontinuities, the phase field formulations have emerged as the most effective methods, but they require solid physical and mathematical foundations, which is typically accomplished with firm mathematical grounding in the corresponding sharp interface model. - Foundations of continuum theories: Mixing, composites and generalized continua. Moving interfaces typically require mass transport, so that mass diffusion models naturally fall within the scope of phase field models. However, the presence of lattice structure in crystalline solids implies fundamental differences between diffusion in fluids and diffusion in solids and thus, different continuum formulations. Higher order continua have been proposed in connection with most problems discussed in this course. - Granular matter is the second most manipulated type of matter after water, with a range of applications, including soils, powders, colloids, amorphous metals and alloys. Depending on the state, they exhibit solid-like or liquid-like properties. Microscopically, granular materials are strongly disordered and require the identification of new variables and descriptors to relate rheological features such as dilatancy (volume change under shear) or strain localization to vortex flow patterns and force chains.  The course is addressed to young researchers including doctoral students, postdocs and early career faculty. Upon completion of the course, attendees will acquire deeper understanding of the following questions. - How far have the understanding and mesoscale modeling advanced in recent decades? - What are the key open questions that require further research? - What are the mathematical and physical requirements for a mesoscale model intended to provide either insight or a predictive engineering tool.