This volume, "Seminar on the Riemann Problem, Complete integrability and Arithmetic Applications", contains a series of lectures presented at a seminar of the same title given by D.and G. Chudnovsky and held in 1979-1980 at the Institute des Hautes Etudes Scientifiaues (IHES) in Bures-sur-Yvette, France (1979),'and at Columbia University in the City of New York, U.S.A. The Seminar speakers examine different aspects of analytic and arithmetic problems arising in various ways from contemporary studies of the Riemann boundary value and monodromy problems. Particular subdivisions of the volume are the following: studies in spectral theory and completely integrable systems (inverse scattering method); the Riemann monodromy problem and statistical mechanics; Pade approximations associated with the Riemann boundary value problem and arithmetical applications to transcendental numbers.