We are concerned with the solution of linear operator equations with a compact operator. These operators do not have a bounded inverse and therefore thes equations have to be regularized before solution. The Arnoldi process provides a convenient way to reduce a compact operator to a nearby operator of finite rank and we regularize with Tikhonov's method. This talk discusses properties of this simple solution approach. This is joint work with M. Kuian and R. Ramlau.