This volume aims to disseminate a number of new ideas that have emerged in the last few years in the field of numerical simulation, all bearing the common denominator of the multiscale or multilevel paradigm. This covers the presence of multiple relevant scales in a physical phenomenon; the detection and representation of structures, localized in space or in frequency, in the solution of a mathematical model; the decomposition of a function into details that can be organized and accessed in decreasing order of importance; and the iterative solution of systems of linear algebraic equations using multilevel decompositions of finite dimensional spaces.