Abstract: In this article, we give a survey of some progress of the complex geometry, mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years. In particular, we explore some works in the special area in Dierential Geometry, Lie Group and Complex Homogeneous Space. Together with the special area in nonlinear analysis on complex manifolds, they are the two major aspects of my research interests.

Key words: Invariant structure, Homogeneous space, Complex torus bundles, Hermitian manifolds, Reductive Lie group, Compact manifolds, Ricci form, Locally conformal Kahler manifolds