Beginning with an introduction to the T-product approach in the theory of a particle interacting with bosonic fields as applied, for example, to the linearized polaron model, the book goes on to deal with the equilibrium state objective being to derive Bogolubov's inequality for the reduced free energy of the polaron. The third chapter deals with some problems related to the non-equilibrium polaron theory, including polaron kinetics. Finally, alternative methods used in polaron theory are also presented and compared with Bogolubov's method.