In the talk we discuss a construction of a large family of complex manifolds. This family includes Hopf and Calabi-Eckmann manifolds. Although the question whether a given differentiable manifold admits a complex structure is extremely hard, our construction completely answers it in the case of manifolds with "large" group of U(1)^m-symmetries. We discuss several very basic geometric characteristics  of these manifolds including Dolbeault cohomology and field of meromorphic functions.