  Let X be a closed Riemannian manifold and let H\hookrightarrow X be an
embedded hypersurface. Let X=X_+ \cup_H X_- be a decomposition of X into two
manifolds with boundary, with X_+ \cap X_- = H. In this expository article,
surgery -- or gluing -- formul\ae for several geometric and spectral invariants
associated to a Dirac-type operator \eth_X on X are presented. Considered in
detail are: the index of \eth_X, the index bundle and the determinant bundle
associated to a family of such operators, the eta invariant and the analytic
torsion. In each case the precise form of the surgery theorems, as well as the
different techniques used to prove them, are surveyed.
