Chapter 10 is devoted to the extension theory for a symmetric operator A with gaps. It is shown that the class of selfadjoint extensions of A preserving a gap \((\alpha ,\beta )\) contains two extremal extensions: \(\widehat {A}_\alpha \) and \(\widehat A_\beta \) . In Sect. 10.1 in the framework of appropriate boundary triples the extremal extensions of A are characterized in terms of the limit values of the corresponding Weyl function \(M(\cdot )\) . Extremal properties of direct sums of extensions with a gap are studied. In Sect. 10.2 the left and right gap-preserving properties for a symmetric operator A with a gap are introduced and some criteria for A to have such a gap-preserving property are found. In Sect. 10.3 we describe selfadjoint extensions \(\widetilde A\) of A that insert a prescribed finite number of eigenvalues in the gap \((\alpha ,\beta )\) , as well as extensions \(\widetilde A\) with point spectra within the gap accumulating to one of the endpoints. In Sect. 10.5 the quadratic forms technique is applied to the operator A with a gap to describe the number of eigenvalues of an extension \(\widetilde A\) within the gap. In Sect. 10.7 the Kreı̆n problem regarding the operator A with several gaps is discussed.