Fitting ideals have been instrumental in proofs of many results in arithmetic. In the first part of the talk we explain a result on Fitting ideals of etale K-groups obtained jointly with D. Burns. Actually this is an outgrowth of a proof of some part of the equivariant Tamagawa number conjecture for Tate motives. In the second part we present a much less sophisticated approach to Fitting ideals of Iwasawa modules and hope to explain that this more naive and explicit approach also leads to interesting arithmetical consequences.