Abstract: In this expository talk, we discuss some classical and recent results on holomorphic functions on subdomains of complex projective spaces. By a Theorem of Takeuchi, these domains are of two types : some on which all holomorphic functions are constant, and others on which there are many holomorphic functions. The main problem is to obtain a full geometric characterization of each type. This is related to the work of Lins-Neto, Siu and Shaw on non-existence of smooth Levi-Flat hypersurfaces in Projective space. Some conjectures in this direction are also discussed.