  An asymptotic stability analysis of spatially homogeneous models of Bianchi
type containing tilted perfect fluids is performed. Using the known attractors
for the non-tilted Bianchi type universes, we check whether they are stable
against perturbations with respect to tilted perfect fluids. We perform the
analysis for all Bianchi class B models and the Bianchi type VI_0 model. In
particular, we find that none of the non-tilted equilibrium points are stable
against tilted perfect fluids stiffer than radiation. We also indicate parts of
the phase space where new tilted exact solutions might be found.
