This book evolved from a set of lectures presented under the auspices of the Conference Board of Mathematical Sciences at the Case Institute of Technology in September 1984. The original objective of the lectures was to present an introduction to the theory and applications of $J$ inner matrices. However, in revising the lecture notes for publication, the author began to realize that the spaces ${\mathcal H}(U)$ and ${\mathcal H}(S)$ are ideal tools for treating a large class of matrix interpolation problems including ultimately two-sided tangential problems of both the Nevanlinna-Pick type and the Caratheodory-Fejer type, as well as mixtures of these. Consequently, the lecture notes were revised to bring ${\mathcal H}(U)$ and ${\mathcal H}(S)$ to center stage. This monograph is the first systematic exposition of the use of these spaces for interpolation problems.