  We review the concepts of the index of a Fredholm operator, the spectral flow
of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of
Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of
operators of Dirac type on closed manifolds and manifolds with boundary. We
emphasize various (occasionally overlooked) aspects of rigorous definitions and
explain the quite different stability properties. Moreover, we utilize the heat
equation approach in various settings and show how these topological and
spectral invariants are mutually related in the study of additivity and
nonadditivity properties on partitioned manifolds.
