  The paper concerns $L^1$- convergence to equilibrium for weak solutions of
the spatially homogeneous Boltzmann Equation for soft potentials $(-4\le
\gm<0$), with and without angular cutoff. We prove the time-averaged
$L^1$-convergence to equilibrium for all weak solutions whose initial data have
finite entropy and finite moments up to order greater than $2+|\gm|$. For the
usual $L^1$-convergence we prove that the convergence rate can be controlled
from below by the initial energy tails, and hence, for initial data with long
energy tails, the convergence can be arbitrarily slow. We also show that under
the integrable angular cutoff on the collision kernel with $-1\le \gm<0$, there
are algebraic upper and lower bounds on the rate of $L^1$-convergence to
equilibrium. Our methods of proof are based on entropy inequalities and moment
estimates.
