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Let X be a closed Riemannian manifold and let H !→ X be an embedded hypersurface. Let X = X_{+} ∪_H X_{−} be a decomposition of X into two manifolds with boundary, with X_{+} ∩X_{−} = H. In this expository article, surgery – or gluing – formulæ for several geometric and spectral invariants associated to a Dirac-type operator ðX on X are presented. Considered in detail are: the index of ðX, 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.