We summarize recent attempts to calculate the flavor asymmetry of the nucleons sea quark distributions in the large-$N_c$ limit, where the nucleon can be described as a soliton of an effective chiral theory. We discuss the leading-twist longitudinally polarized and transversity antiquark distributions, $Deltabar u (x) - Deltabar d (x)$ and $deltabar u (x) - deltabar d (x)$, as well as the unpolarized one, $bar u (x) - bar d (x)$, which appears only in the next-to-leading order of the $1/N_c$-expansion. Results for $bar u (x) - bar d (x)$ are in good agreement with the recent Drell-Yan data from the FNAL E866 experiment. The longitudinally polarized antiquark asymmetry, $Deltabar u (x) - Deltabar d (x)$, is found to be larger than the unpolarized one.
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