This volume consists of eight papers on new advances in interpolation theory for matrix functions and completion theory for matrices and operators. Much emphasis is placed on different interpolation and completion problems when the interpolant is estimated in two different norms. The book also focusses on the study of the spectra of different completions of 2 x 2 block matrices when originally all entries are specified except the lower left corner. A third theme concerns two-sided tangential interpolation problems for real rational matrix functions, and also for the time varying case. A tangential moment problem is also analyzed. All papers deal with related problems of modern matrix analysis, operator theory, complex analysis and system theory and will appeal to a wide group of mathematicians and engineers. The material can be used for advance courses and seminars. Contents: Editorial Introduction • D. Alpay/P. Loubaton: The tangential trigonometric moment problem on an interval and related topics • M. Bakonyi/V.G. Kaftal/G. Weiss/H.J. Woerdeman: Maximum entropy and joint norm bounds for operator extensions • J.A. Ball/I. Gohberg/M.A. Kaashoek: Bitangential interpolation for input-output operators of time varying systems: the discrete time case • J.A. Ball/I. Gohberg/L. Rodman: Two-sided tangential interpolation of real rational matrix functions • H. Du/C. Gu: On the spectra of operator completion problems • C. Foias/A.E. Frazho/W.S. Li: The exact H2 estimate for the central Hinterpolant • A.E. Frazho/s.M. Kherat: On mixed H2 - Htangential interpolation • I. Gohberg/C.Gu: On a completion problem for matrices