Chapter 6 is devoted to the theory of Herglotz–Nevanlinna functions ( \({\mathcal R}\) -functions). In Sects. 6.1–6.2 we recall classical integral representations of scalar and operator-valued functions from Herglotz–Nevanlinna, Kats–Aronszajn–Donoghue, and Stieltjes classes. We also recall here the Stieltjes inversion formula. In Sect. 6.3 we discuss generalized Stieltjes classes \(\mathcal S^{\pm \kappa }[{\mathcal H}\,]\) introduced by the authors for description of generalized resolvents corresponding to exist space extensions of a non-negative operator that have \(\kappa \) negative eigenvalues. We present integral representations of operator-valued functions from the classes \(\mathcal S^{\pm \kappa }[{\mathcal H}\,]\) and use them to improve the known results on integral representations of operator-valued functions from the Kreı̆n-Langer classes \({\mathcal N}_{\kappa }^{\pm }[{\mathcal H}\,]\) .