  We study the Wigner function for massive spin-1/2 fermions in electromagnetic
fields. Dirac form kinetic equation and Klein-Gordon form kinetic equation are
obtained for the Wigner function, which are derived from the Dirac equation.
The Wigner function and its kinetic equations are expanded in terms of the
generators of Clifford algebra and a complicated system of partial differential
equations is obtained. We prove that some component equations are
automattically satisfied if the rest ones are fulfilled. In this thesis two
methods are proposed for calculating the Wigner function, which are proved to
be equivalent. The Wigner function is analytically calculated following the
standard second-quantization procedure in the following cases: free fermions
with or without spin imbalance, in constant magnetic field, in constant
electric field, and in constant parallel electromagnetic field. Strong-field
effects, such as the Landau levels and Schwinger pair-production are reproduced
using the Wigner function approach. For an arbitrary space-time dependent field
configuration, a semi-classical expansion with respect to the reduced Planck's
constant $\hbar$ is performed. We derive general expressions for the Wigner
function components at linear order in $\hbar$, in which order the spin
corrections start playing a role. A generalized Bargmann-Michel-Telegdi (BMT)
equation and a generalized Boltzmann equation are obtained for the undetermined
polarization density and net fermion number density, which can be used to
construct spin-hydrodynamics in the future. We also make a comparison between
analytical results and the ones from semi-classical expansion, which shows
coincidence for weak electromagnetic fields and small spin imbalance.
