In this talk we consider iterative Krylov subspace algorithms to solve ill-posed inverse problems with l-p regularization. The main idea is to consider suitable, adaptively-defined preconditioners that allow using the usual 2-norm. The preconditioners can be updated at each step and/or after some iterations have been performed, leading to two different approaches: one is based on the idea of iteratively reweighted least squares, and can be obtained considering flexible preconditioned Krylov subspaces; the second approach is based on restarting the Arnoldi algorithm. Numerical examples are given in order to show the effectiveness of these new methods and comparison with some other existing algorithms are made.