We consider the generation of magnetic fields under astrophysical conditions. Principal attention is paid to ``dynamo'' mechanisms, i.e., mechanisms in which the energy of the magnetic field is drawn from the kinetic energy of plasma motion. The important role played by turbulent dynamo mechanisms is emphasized. The dynamo problem itself is divided into two: 1) generation of regular magnetic fields (i.e., the dynamo mechanism of fields having scales commensurate with the cosmic object itself by turbulent pulsations); 2) generation of random fields. A review is presented of the existing theories for the generation of regular fields (reference is made, in particular, to the work of Steenbeck and co-workers and of Parker), and a generalization of the existing results to include large magnetic Reynolds number Rem, characteristic of cosmic plasma, is indicated. Astrophysical examples are given. The existing theories in the dynamics of random fields are also reviewed. Results are presented on the turbulent dynamo in the presence of acoustic turbulence. Analogies with the question of excitation of vortices in a field of acoustic turbulence is indicated. The question of the turbulent dynamo in the field of ``Kolmogorov'' turbulence is discussed. Finally, an essential problem is that of the steady-state field produced by a nonlinear effect. It is shown that in the presence of a non-weak field there appears a gyrotropy acting in opposition to the action of the usual rotational gyrotropy. This observed gyrotropy is called magnetic and can cause nonlinear stabilization of the magnetic field.