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Greenbergs first conjecture(GC)affirms that for any totally real field K,the associated unramified Iwasawa module X(K∞)is finite.It is closely related to the Vandiver conjecture,which predicts the vanishing of X(K∞)for K = Q(μp)+.The Generalized Greenberg Conjecture(GGC)claims that for any number field K,the unramified Iwasawa module X((K))should be pseudo-null over the Iwasawa algebra of (K)/K,where (K) is the conposite of all Zp-extensions of K.In this talk we aim to concentrate on the significance and depth of Greenbergs conjectures in Iwasawa theory: the implication from GC to the Main Conjecture for K∞/K,and the formulation through GGC of a Main conjecture for (K)/K interms of higher Chern classes and characteristic symbols.