  Fast radio bursts (FRBs) are observed to be highly polarized. Most have high
linear polarization but a small fraction show significant circular
polarization. We systematically investigate a variety of polarization
mechanisms of FRBs within the magnetar theoretical framework considering two
emission sites inside and outside the magnetosphere. For each site, we discuss
both intrinsic radiation mechanisms and propagation effects. Inside the
magnetosphere, we investigate the polarization properties of both coherent
curvature radiation and inverse Compton scattering by charged bunches and
conclude that both mechanisms produce 100\% linear polarization at an on-axis
geometry but can produce circular polarization if the viewing angle is off
axis. The lack of circular polarization for the majority of bursts requires
that the bunches have a large transverse dimension size. Resonant cyclotron
absorption within magnetosphere may produce high circular polarization if
electrons and positrons have an asymmetric Lorentz factor distribution. Outside
the magnetosphere, the synchrotron maser emission mechanism in general produces
highly linearly polarized emission. Circular polarization would appear at
off-beam angles but the flux is greatly degraded and such bursts are not
detectable at cosmological distances. Synchrotron absorption in a nebula with
ordered magnetic field may reduce the circular polarization degree. Cyclotron
absorption in a strongly magnetized medium may generate significant circular
polarization. Faraday conversion in a medium with field reversal can convert
one polarization mode to another. The two absorption processes require
stringent physical conditions. Significant Faraday conversion may be realized
in a magnetized dense environment involving binary systems or supernova
remnants.
