  We discuss the cosmological evolution of matter sources with small
anisotropic pressures. This includes electric and magnetic fields,
collisionless relativistic particles, gravitons, antisymmetric axion fields in
low-energy string cosmologies, spatial curvature anisotropies, and stresses
arising from simple topological defects. The COBE microwave sky maps are used
to place strong limits on the possible contribution of these sources to the
total density of the universe. We explain why the limits obtained from
primordial nucleosynthesis are generally weaker than those imposed by the
microwave background isotropy. The effect of inflation on all these stresses is
also calculated.
