  We develop a model based on 3D mean-field MHD for the generation of large
scale magnetic fields in fully convective objects like low-mass stars, brown
dwarfs and possibly gaseous planets. The dynamo process is of alpha^2 type and
thus differs from the shell-dynamo at work in more massive stars.The alpha^2
dynamo becomes supercritical for Rossby numbers Ro\la 10. It generates a
large-scale, non-axisymmetric, steady field that is symmetric with respect to
the equatorial plane. Saturation of the alpha^2-generated field at the
equipartition yields strengths of several kGauss, in agreement with
observations of active M dwarfs, and provides a qualitative explanation for the
observed activity saturation in late M stars. For brown dwarfs with a
conductive core, as occurs at the center of the most massive and oldest of
these objects, we have also studied an alpha^2-Omega dynamo, i.e. the effect of
differential rotation. In this case the field is predominantly toroidal,
axisymmetric and oscillatory, like the solar field. The topology of the field
in the fully convective objects exhibits a high order multipole character that
differs from the aligned dipole field generated by the alpha-Omega dynamo. The
strong reduction of the dipolar component due to the field non-axisymmetry
should considerably reduce the Alfven radius and thus the efficiency of
magnetic braking, providing an appealing explanation for the decreasing angular
momentum loss rate observed in low-mass stars and brown dwarfs. This may have
also implications for cataclysmic variables below the period gap. In spite of
this large-scale field, the decreasing conductivity in the dominantly neutral
atmosphere of these objects prevents the current generation necessary to
support a chromosphere and thus activity. (Abridged)
