I. Analysis of the Caratheodory Interpolation Problem -- II. Analysis of the Caratheodory Interpolation Problem for Positive-Real Functions -- III. Schur Numbers, Geophysics and Inverse Scattering Problems -- IV. Contractive Expansions on Euclidian and Hilbert Space -- V. Contractive One Step Intertwining Liftings -- VI. Isometric and Unitary Dilations -- VII. The Commutant Lifting Theorem -- VIII. Geometric Applications of the Commutant lifting Theorem -- IX. H? Optimization and Functional Models -- X. Some Classical Interpolation Problems -- XI. Interpolation as a Momentum Problem -- XII. Numerical Algorithms for H? Optimization in Control Theory -- XIII. Inverse Scattering Algorithms for the Commutant Lifting Theorem -- XIV. The Schur Representation -- XV. A Geometric Approach to Positive Definite Sequences -- XVI. Positive Definite Block Matrices -- XVII. A Physical Basis for the Layered Medium Model -- References -- Notation