This chapter explores the theoretical foundations of beam-plasma instabilities. It starts by introducing the kinetic theory of plasmas using the Vlasov-Maxwell equations, which form the basis for analyzing instabilities. The chapter details the electrostatic beam-plasma instability, showing that for blazar-induced pair beams, oblique modes in the kinetic regime dominate. This chapter concludes by examining key non-linear processes, including nonlinear Landau damping and modulation instability, which influence the instability’s evolution and potential energy dissipation into the IGM.