  We introduce and study the class of almost limited sets in Banach lattices,
that is, sets on which every disjoint weak$^{*}$ null sequence of functionals
converges uniformly to zero. It is established that a Banach lattice has order
continuous norm if and only if almost limited sets and $L$-weakly compact sets
coincide. In particular, in terms of almost Dunford-Pettis operators into
$c_{0}$, we give an operator characterization of those $\sigma$-Dedekind
complete Banach lattices whose relatively weakly compact sets are almost
limited, that is, for a $\sigma$-Dedekind Banach lattice $E$, every relatively
weakly compact set in $E$ is almost limited if and only if every continuous
linear operator $T:E\rightarrow c_{0}$ is an almost Dunford-Pettis operator.
