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Classical Dirac-Schrödinger operators are given by Dirac-type operators on a smooth manifold, together with a potential. I will describe a general notion of Dirac-Schrödinger operators with arbitrary signatures (with or without gradings), which allows us to study index pairings and spectral flow simultaneously. I will then provide a general Callias Theorem, which computes the index (or the spectral flow) of Dirac-Schrödinger operators on a compact hypersurface. In a special case, I will explain how the result can be computed in terms of well-known index pairings on the hypersurface.