This paper discusses Iitaka's theory of quasi-Albanese maps and provides a detailed proof of Kawamata's theorem on quasi-Albanese maps for varieties with logarithmic Kodaira dimension zero. The theory relies on Deligne's mixed Hodge theory for smooth algebraic varieties. The paper is structured into sections covering preliminaries, properties of quasi-abelian varieties, characterizations of abelian varieties and complex tori, subadditivity of the logarithmic Kodaira dimension, semipositivity theorems, weak positivity theorems, finite covers of quasi-abelian varieties, and the proof of Kawamata's theorem. The main result is Theorem 1.1, which states that for a smooth algebraic variety X over C, there exists a morphism α: X → A to a quasi-abelian variety A such that any other morphism β: X → B to a quasi-abelian variety B factors through α. The paper also proves Theorem 1.2, which shows that the quasi-Albanese map α: X → A is dominant with irreducible general fibers for varieties with logarithmic Kodaira dimension zero. Additional results include Corollaries 1.4 and 1.5, which provide conditions under which the quasi-Albanese map is birational or has certain properties. The paper also discusses the relationship between quasi-abelian varieties and abelian varieties, and provides a detailed proof of the existence of quasi-Albanese maps and varieties. The paper concludes with a summary of the key results and their implications for the theory of quasi-Albanese maps.This paper discusses Iitaka's theory of quasi-Albanese maps and provides a detailed proof of Kawamata's theorem on quasi-Albanese maps for varieties with logarithmic Kodaira dimension zero. The theory relies on Deligne's mixed Hodge theory for smooth algebraic varieties. The paper is structured into sections covering preliminaries, properties of quasi-abelian varieties, characterizations of abelian varieties and complex tori, subadditivity of the logarithmic Kodaira dimension, semipositivity theorems, weak positivity theorems, finite covers of quasi-abelian varieties, and the proof of Kawamata's theorem. The main result is Theorem 1.1, which states that for a smooth algebraic variety X over C, there exists a morphism α: X → A to a quasi-abelian variety A such that any other morphism β: X → B to a quasi-abelian variety B factors through α. The paper also proves Theorem 1.2, which shows that the quasi-Albanese map α: X → A is dominant with irreducible general fibers for varieties with logarithmic Kodaira dimension zero. Additional results include Corollaries 1.4 and 1.5, which provide conditions under which the quasi-Albanese map is birational or has certain properties. The paper also discusses the relationship between quasi-abelian varieties and abelian varieties, and provides a detailed proof of the existence of quasi-Albanese maps and varieties. The paper concludes with a summary of the key results and their implications for the theory of quasi-Albanese maps.