  We first derive the energy dispersion of bilayer MoS$_{2}$ in the presence of
a perpendicular electric field $E_z$. We show that the band gap and layer
splitting can be controlled by the field $E_z$. Away from the $k$ point, the
intrinsic SOC splitting increases in the conduction band but is weakly affected
in the valence band. We then analyze the band structure in the presence of a
perpendicular magnetic field $B$ and the field $E_z$, including spin and valley
Zeeman terms, and evaluate the Hall and longitudinal conductivities. We discuss
the numerical results as functions of the fields $B$ and $E_z$ for finite
temperatures. The field $B$ gives rise to a significant spin splitting in the
conduction band, to a beating in the Shubnikov-de Haas (SdH) oscillations when
it's weak, and to their splitting when it's strong. The Zeeman terms and
$E_{z}$ suppress the beating and change the positions of the beating nodes of
the SdH oscillations at low $B$ fields and enhance their splitting at high $B$
fields. Similar beating patterns are observed in the spin and valley
polarizations at low $B$ fields. Interestingly, a $90\%$ spin polarization and
a $100\%$ square-wave-shaped valley polarization are observed at high $B$
fields. The Hall-plateau sequence depends on $E_z$. These findings may be
pertinent to future spintronic and valleytronic devices.
