Abstract: For a Z_p-tower of global fields, Iwasawa's classical theorem says that the exponent of p in the n-th class number of the tower is given by a linear polynomial in p^n and n for large n. A fundamental and wide open problem is whether the polynomial stability for the the exponent of p in the n-th class number remains true for a p-adic Lie tower of global fields. In this talk, we shall explain some new results suggesting that the answer should be positive, including a proof of the abelian case over function fields (Greenberg's conjecture).