  It is known that, for a positive Dunford-Schwartz operator in a
noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a
noncommutative Orlicz space with order continuous norm, the corresponding
ergodic averages converge bilaterally almost uniformly. We show that these
averages converge almost uniformly in each noncommutative symmetric space $E$
such that $\mu_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $\mu_t(x)$
is a non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type
multiparameter ergodic theorems are studied. A wide range of noncommutative
symmetric spaces for which Dunford-Schwartz-type individual ergodic theorems
hold is outlined. Also, almost uniform convergence in noncommutative
Wiener-Wintner theorem is proved.
