For closed manifolds, there is a highly elaborated theory of number-valued invariants, attached to the underlying manifold, structures and differential operators. On open manifolds, nearly all of this fails, with the exception of some special classes. The goal of this monograph is to establish for open manifolds, structures and differential operators an applicable theory of number-valued relative invariants. This is of great use in the theory of moduli spaces for nonlinear partial differential equations and mathematical physics. The book is self-contained: in particular, it contains an outline of the necessary tools from nonlinear Sobolev analysis.Contents:Absolute Invariants for Open Manifolds and BundlesNon-linear Sobolev StructuresThe Heat Kernel of Generalized Dirac OperatorsTrace Class PropertiesRelative Index TheoryRelative ζ–Functions, η–Functions, Determinants and TorsionScattering Theory for Manifolds with Injectivity Radius ZeroReadership: Graduate students, mathematicians and physicists interested in global analysis.