In this talk, we present recent results obtained in collaboration with Shin-ichi Matsumura, Juanyong Wang, and Qimin Zhang, concerning compact Kähler manifolds whose anticanonical bundle is nef. It has been conjectured that every compact Kähler manifold with nef anticanonical bundle admits a locally trivial fibration over a Calabi–Yau manifold. Significant progress has been made in the projective setting, in particular through the works of Cao and Höring, who established the structure of projective varieties with nef anticanonical bundles. However, extending these results to the compact Kähler setting poses substantial challenges. The recent breakthrough of Wenhao Ou, who resolved a conjecture proposed by Boucksom, Demailly, Păun, and Peternell, provides new tools to address these difficulties. Building on these developments, we aim to show that the conjecture holds in full generality for compact Kähler manifolds.