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In this article we study the structure of klt projective varieties with nefanticanonical divisor (and more generally, varieties of semi-Fano type),especially the canonical fibrations associated to them. We show that: 1. theAlbanese map for such variety is a locally constant fibration (that is, ananalytic fibre bundle with connected fibres that $X$ is equal to the product ofthe universal cover of the Albanese torus by the fibre of the Albanese mapquotient by a diagonal action of the fundamenatl group of the Albanese torus);2. if the smooth locus is simply connected, the MRC fibration of such varietyis an everywhere defined morphism and induces a decomposition into a product ofa rationally connected variety and of a projective variety with trivialcanonical divisor. These generalize the corresponding results for smoothprojective varieties with nef anticanonical bundle in Cao (2019) andCao-H\"oring (2019) to the klt case, and can be also regarded as a partialextension of the singular Beauville-Bogomolov decomposition theorem proved bysuccessive works of Greb-Kebekus-Peternell (2016), Druel (2018),Guenencia-Greb-Kebekus (2019) and H\"oring-Peternell (2019).

Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities