We discuss a dynamo mechanism driven by weak plasma turbulence and show that turbulent plasma waves may generate and maintain cosmic magnetic field. A dynamo equation is derived from the magnetic induction equation based on mean field electrodynamics. In the usual α–ω dynamo theory, the source term in the dynamo equation arises from α–effect associated with the convective motion of the fluid. In contrast, in our theory the source term is determined by “P–effcct” associated with weakly turbulent waves (e.g. Alfvén waves) in the plasma. We suggest that “P–ω” dynamo may be operative either in the presence or absence of convection. The sole requirement for its operation is the existence of weak plasma turbulence in the source region of the cosmic magnetic field.