In Chap. 12 we adapt the boundary triple approach from Chaps. 8, 9 and 10 to the case of a non-densely defined symmetric operator A acting in a Hilbert space \(\mathfrak H\) . Section 12.1 recalls the von Neumann–Krasnoselskii extension theory for a non-densely defined symmetric operator A. In Sect. 12.2 the notions of the boundary triple and the Weyl function are adapted to the linear relations framework and used for description of proper extensions of A and their spectra. A coupling construction for two boundary triples is discussed. In Sect. 12.3 the concepts of B-generalized and double B-generalized boundary triples, the corresponding Weyl functions, and Kreı̆n type resolvent formula are discussed. In Sect. 12.4 we present two proofs of the Naı̆mark resolvent formula—one based on the Naı̆mark dilation theorem and the other using the coupling construction from Sect. 12.2. Moreover, we discuss here the Kreı̆n–Naı̆mark formula for generalized resolvents of Hermitian contractions, non-negative operators, or symmetric operators with a gap. In Sects. 12.5–12.6 the theory of \({\mathfrak L}\) -resolvent and \({\mathfrak L}\) -preresolvent matrices for a symmetric operator A with a proper gauge \({\mathfrak L}\subset {\mathfrak H}\) is exhibited. It is shown that the \({\mathfrak L}\) -preresolvent matrix of A coincides with the Weyl function of an appropriate boundary triple of the restriction \(S = A \upharpoonright {\mathfrak L}^{\perp }\) of A. Moreover, the \({\mathfrak L}\) -resolvent matrix is explicitly expressed by means of boundary mappings and the abstract analog of polynomials of the first and second kind. As an application of this theory, the Hamburger, Hausdorff and Stieltjes moment problems, as well as the moment problem with either one or several gaps, are considered. We also discuss the Schur algorithm, the factorization of the resolvent matrix and the extremal solutions of the Stieltjes moment problem.