This book is devoted primarily to topics in interpolation for scalar, matrix and operator valued functions. About half the papers are based on lectures which were delivered at a conference held at Leipzig University in August 1994 to commemorate the 80th anniversary of the birth of Vladimir Petrovich Potapov. The volume also contains the English translation of several papers relatively unknown in the West, two expository papers written especially for this volume, and historical material based on reminiscences of former colleagues, students and associates of Mr. Potapov. Several examples of interpolation problems of the Nevanlinna-Pick and Caratheodory-Fejer type are included as well as moment problems and problems of integral representation in several settings. The major themes covered include applications of the Potapov method of fundamental matrix inequalities, multiplicative decompositions of J-inner matrix valued functions, the abstract interpolation problem, canonical systems of differential equations and interpolation in spaces with an indefinite metric, and this text should appeal to mathematicians specializing in pure and applied mathematics and engineers who work in systems theory and control.