  There are several astrophysical situations where one needs to study the
dynamics of magnetic flux in partially ionized turbulent plasmas. In a
partially ionized plasma the magnetic induction is subjected to the ambipolar
diffusion and the Hall effect in addition to the usual resistive dissipation.
In this paper we initiate the study of the kinematic dynamo in a partially
ionized turbulent plasma. The Hall effect arises from the treatment of the
electrons and the ions as two separate fluids and the ambipolar diffusion due
to the inclusion of neutrals as the third fluid. It is shown that these
nonideal effects modify the so called $\alpha$ effect and the turbulent
diffusion coefficient $\beta$ in a rather substantial way. The Hall effect may
enhance or quench the dynamo action altogether. The ambipolar diffusion brings
in an $\alpha$ which depends on the mean magnetic field. The new correlations
embodying the coupling of the charged fluids and the neutral fluid appear in a
decisive manner. The turbulence is necessarily magnetohydrodynamic with new
spatial and time scales. The nature of the new correlations is demonstrated by
taking the Alfv\'enic turbulence as an example.
