This chapter is dedicated to the construction of triangular and functional models of linear bounded non-self-adjoint operators A acting on a separable Hilbert space H. (Note that all the Hilbert spaces we consider herein are separable.) The “defect” space E is assigned to every such operator A (generally speaking, this space is equivalent to \( \overline{A_IH} \) , where \( {A}_I=\frac{1}{2i}\left(A-{A}^{\ast}\right) \) ). All further developments, per se, take place in the “restriction” of the analytic structures from the space H to this space E (for instance, the resolvent (A − λI)−1). The proposed approach is based on the symbiosis of the following ideas and methods: Livšic’s theory of characteristic functions; Nagy–Foias dilation theory; and Lax–Phillips geometrical scattering theory. First of all, we introduce the notion of a local colligation Δ, which is a natural logical construction in the study of the degree of deviation from self-adjointness, \( {A}_I=\frac{1}{2i}\left(A-{A}^{\ast}\right) \) , of the operator A. Then, using it, we construct an open system FΔ. It is essential that FΔ not only describes the non-unitarity of the semigroup Zt =  exp {itA}, but it is also a “window” through which energy flows in and out if we consider Zt as an evolutionary operator. This open system FΔ with its conservation law lies at the basis of the construction of the unitary dilation Ut in ℋ of the semigroup Zt