ζ-function regularized traces can be used as a tool, in infinite-dimensional geometry, to define analogs of usual geometric structures (e.g. symplectic or Dirac structures) in the infinite-dimensional setting, and to build topological invariants. In this talk we will introduce the analytic setting needed in order to define the ζ-function regularization of traces and such tools will be used to define geometric structures on infinite-dimensional Lie algebras and infinite-dimensional manifolds.