  The quark and gluon viscosities are calculated in quark-gluon plasmas to
leading orders in the coupling constant by including screening. For weakly
interaction QCD and QED plasmas dynamical screening of transverse interactions
and Debye screening of longitudinal interactions controls the infrared
divergences. For strongly interacting plasmas other screening mechanisms taken
from lattice calculations are employed. By solving the Boltzmann equation for
quarks and gluons including screening the viscosity is calculated to leading
orders in the coupling constant. The leading logarithmic order is calculated
exactly by a full variational treatment. The next to leading orders are found
to be very important for sizable coupling constants as those relevant for the
transport properties relevant for quark-gluon plasmas created in relativistic
heavy ion collisions and the early universe.
