This basic three-body interaction is phenomenologically very rich, and covers several diAerent processes, from stochastic acceleration to nonlinear Compton scattering. We discuss both classical and quantum descriptions of this interaction, and illustrate their physical consequences. In the classical description, we use a Superhamiltonian formulation of the particle motion (electron, or ion) in two electromagnetic waves, and show that it is non integrable. DiAusion in phase-space describes stochastic acceleration, and can explain the broad electron energy spectrum observed in intense laser-plasma interactions. In the non-relativistic limit, it can also be explain the stochastic heating of ions by electrostatic waves. In the quantum description, we use Volkov solutions of the relativistic wave equations (Klein-Gordon and Dirac), and shown that they are able to explain the nonlinear Compton scattering of photons and plasmons, thus generalizing the traditional view of Compton scattering, which only involves photons. This problem is highly relevant to in the present experiments with PetaWatt laser systems, where the nonlinear quantum plasma regime becomes accessible. Compton scattering is the basic ingredient for a consistent plasma quantum theory. It is also known that the so-called inverse Compton scattering, which corresponds to scattering of low energy photons by highly relativistic particles, is very important in Astrophysics. The same Volkov solutions can also be used to describe the nonlinear regime of quantum Landau damping, showing that in this regime the electrons can emit and absorb more than one plasmon. Explicit expressions for this multi-plasmon Landau damping can be derived. This explains the possible occurrence of multi-plasmon absorption of electrostatic waves by electrons, and confirms previous simulation results. A similar formalism can be used to describe photon Landau damping of electron plasma waves. Such a similarity with quantum electron Landau damping is not surprising, given the undulatory nature of photons. Relevance of both processes to laser acceleration is exemplified.