The author studies principal bundles \(\pi:X \to M\) over a compact complex manifold \(M\) whose structure group is a compact complex torus \(T=\mathbb{C}^ n/ \Lambda\). In general \(X\) is not Kähler even if \(M\) is. Typical examples are Hopf and Calabi-Eckmann manifolds diffeomorphic to products of spheres. These are principal bundles over a product of projective spaces, the fibre is an elliptic curve. The author often assumes that \(H^ 2(M)\) has a Hodge decomposition. In general he defines a characteristic class \(c^ \mathbb{Z} \in H^ 2(M,\Lambda)\) and invariants \(\varepsilon:H^{0,1}_ T \to H^{0,2}_ M\) and \(\gamma:H^{1,0}_ T \to H^{1,1}_ M\). He computes them from \(c^ \mathbb{Z}\). They determine the \(d_ 2\) differentials of the Leray and Borel spectral sequences converging to \(H^ \bullet (X,\mathbb{C})\) and \(H_ X^{\bullet,\bullet}\) respectively, with a variant computing \(H^ \bullet (\Theta_ X)\). Here all these spectral sequences degenerate on \(E_ 3\)-level. The author finds many interesting examples, among them complex structures with different Kodaira dimension on the same \(C^ \infty\) manifold. He also describes the infinitesimal deformations of \(X\) using \(\varepsilon\) and \(\gamma\).