  For a Dunford-Schwartz operator in the $L^p-$space, $1\leq p< \infty$ , of an
arbitrary measure space, we prove pointwise convergence of the conventional and
Besicovitch weighted ergodic averages. Pointwise convergence of various types
of ergodic averages in fully symmetric spaces of measurable functions with
non-trivial Boyd indices is studied. In particular, it is shown that for such
spaces Bourgain's Return Times theorem is valid.
