Free and interacting quantum fields : first order processes on electromagnetic and gravitational backgrounds Modern physics is characterized by two great theories which make it fundamentally different from its predecessor: quantum theory and theory of relativity. In this book we want to bring to the readers attention several solutions to problems connected to the quantum relativistic interaction of particles. Remarkably such solutions furnished rigorous and pertinent explanations of a large set of phenomena both in microscopic world and galactic universe Classical and quantum free fields The gravitational transmutation hypothesis Tomonaga Schwinger representation of dynamics of quantum physical systems Elements of gauge theories The gravitational field interacting with other fields Interaction of scalar spinorial spin vectorial and tensorial particles and the gravitational field described by the Schwarzschild metric Scattering of electrons and photogeneration of gravitons in external gravitational field Interaction of scalar spinorial vectorial spin vectorial and tensorial particles with an axially symmetric gravitational field Software package for anaytical calculation of differential cross sections of gravitational scattering of high spin particles