This chapter shows how the spectral functions appear in physical contexts and proves some of their basic properties. The physical context chosen is quantum field theory and quantum mechanics under external conditions. The chapter expresses relevant physical properties, encoded for example in functional determinants, ground-state energies and heat kernel coefficients, through zeta functions. It explains various relations that depend crucially on the properties of the spectral functions involved and derives needed results using pseudo-differential operator calculus. The chapter describes in detail the differences occurring for elliptic operators on manifolds without and with boundaries. It also explains some peculiarities for global boundary conditions as compared to local ones.