Chapter 8 is devoted to the boundary triple approach to the extension theory of symmetric operators A in Hilbert spaces. The family of proper extensions of a symmetric operator A with equal defect numbers is parameterized by the set of linear relations in a boundary space. The main analytical tool of the investigation of boundary value problems in the framework of boundary triple approach is the abstract Weyl function of A corresponding to a boundary triple. It is introduced and investigated in Sect. 8.4. In particular, the spectrum of proper extension \(\widetilde A\) is characterized via the Weyl function \(M(\cdot )\) and the boundary operator B that corresponds to the extension \(\widetilde A\) . In Sect. 8.5 with each boundary triple it is naturally associated the (unique) Kreı̆n’s formula for canonical resolvents which gives another parametrization of the set of all proper extensions. Both parametrizations make it possible to apply Kreı̆n’s formula to boundary value problems. In Sect. 8.6 boundary triples and Weyl functions for intermediate symmetric extensions of A are calculated and the coupling construction for two symmetric operators is introduced and discussed. Boundary triples and their Weyl functions for different differential operators including Dirac-type operators, Sturm-Liouville operators, 2n-th order differential operators are presented in Sect. 8.7. Eigenvalue interlacing properties of proper extensions of a symmetric operator A with a gap are discussed in Sect. 8.8. In Sect. 8.10 we introduce a more general definition of a boundary triple for an arbitrary closed operator T on a Hilbert space \(\mathfrak H\) and present two statements on such triples that model the situation arising in applications to differential operators.