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 main results include: 1. **Theorem 1.1**: For a smooth algebraic variety \(X\) defined over \(\mathbb{C}\), there exists a morphism \(\alpha: X \to A\) to a quasi-abelian variety \(A\) such that for any other morphism \(\beta: X \to B\) to a quasi-abelian variety \(B\), there is a unique morphism \(f: A \to B\) such that \(\beta = f \circ \alpha\). 2. **Theorem 1.2**: For a smooth variety \(X\) with logarithmic Kodaira dimension zero, the quasi-Albanese map \(\alpha: X \to A\) is dominant and has irreducible general fibers. 3. **Theorem 1.3**: For a smooth variety \(X\) with \(\overline{\kappa}(X) = 0\) and logarithmic irregularity \(\overline{q}(X) = \dim X\), the quasi-Albanese map \(\alpha: X \to A\) is birational, and there exists a closed subset \(Z\) of \(A\) with \(\operatorname{codim}_AZ \geq 2\) such that \(\alpha\) is an isomorphism on \(X \setminus \alpha^{-1}(Z)\) and \(\alpha^{-1}(Z)\) is of pure codimension one. 4. **Corollaries 1.4 and 1.5**: Corollary 1.4 states that a smooth affine variety \(X\) is isomorphic to \(\mathbb{G}_m^n\) if and only if \(\overline{\kappa}(X) = 0\) and \(\overline{q}(X) = n\). Corollary 1.5 states that for a nonempty Zariski open set \(X\) of a quasi-abelian variety \(A\), \(\overline{\kappa}(X) = 0\) if and only if \(\operatorname{codim}_A(A \setminus X) \geq 2\). The paper also includes historical notes, preliminaries on logarithmic Kodaira dimensions and quasi-abelian varieties, and detailed proofs of theorems and corollaries. It aims to make Iitaka's theory of quasi-Albanese maps and Kawamata's result accessible by providing detailed explanations and supplementary comments.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 main results include: 1. **Theorem 1.1**: For a smooth algebraic variety \(X\) defined over \(\mathbb{C}\), there exists a morphism \(\alpha: X \to A\) to a quasi-abelian variety \(A\) such that for any other morphism \(\beta: X \to B\) to a quasi-abelian variety \(B\), there is a unique morphism \(f: A \to B\) such that \(\beta = f \circ \alpha\). 2. **Theorem 1.2**: For a smooth variety \(X\) with logarithmic Kodaira dimension zero, the quasi-Albanese map \(\alpha: X \to A\) is dominant and has irreducible general fibers. 3. **Theorem 1.3**: For a smooth variety \(X\) with \(\overline{\kappa}(X) = 0\) and logarithmic irregularity \(\overline{q}(X) = \dim X\), the quasi-Albanese map \(\alpha: X \to A\) is birational, and there exists a closed subset \(Z\) of \(A\) with \(\operatorname{codim}_AZ \geq 2\) such that \(\alpha\) is an isomorphism on \(X \setminus \alpha^{-1}(Z)\) and \(\alpha^{-1}(Z)\) is of pure codimension one. 4. **Corollaries 1.4 and 1.5**: Corollary 1.4 states that a smooth affine variety \(X\) is isomorphic to \(\mathbb{G}_m^n\) if and only if \(\overline{\kappa}(X) = 0\) and \(\overline{q}(X) = n\). Corollary 1.5 states that for a nonempty Zariski open set \(X\) of a quasi-abelian variety \(A\), \(\overline{\kappa}(X) = 0\) if and only if \(\operatorname{codim}_A(A \setminus X) \geq 2\). The paper also includes historical notes, preliminaries on logarithmic Kodaira dimensions and quasi-abelian varieties, and detailed proofs of theorems and corollaries. It aims to make Iitaka's theory of quasi-Albanese maps and Kawamata's result accessible by providing detailed explanations and supplementary comments.