This book presents a wide range of problems connected with rational approximations of numbers and analytic functions. These problems touch on many topics in contemporary analysis, such as analytic functions, orthogonal polynomials, spectral theory of operators, and potential theory. Motivated by the development of the theory of Pade approximants and the current application of this theory in related disciplines, the authors present an introduction to this circle of ideas. The book is intended for students and future researchers interested in function theory and number theory.