In many areas of science and engineering one is interested in determining the cause of an observed effect. This gives rise to linear inverse problems. The solution of this kind of problems is very sensitive to perturbations in the data. The perturbations may be caused, e.g., by measurement errors as well as round-off errors introduced during the computations. To reduce the sensitivity of the computed solution to perturbations in the data, one employs regularization. Instead of solving the original problem, one may solve an ℓ2-ℓq minimization problem, i.e., one minimizes a weighted sum of a squared Euclidean norm of a fidelity term and the qth power of the ℓq-norm with 0 < q ≤ 2 of a regularization term, where we note that the “ℓq-norm” does not satisfy all properties of a norm for 0 < q < 1. This paper describes an iterated variant of this regularization approach. It is known that iterated variants of Tikhonov regularization yield computed solutions of higher quality than “standard” Tikhonov regularization. We show that iterated ℓ2-ℓq minimization gives computed solutions of higher quality than standard ℓ2-ℓq minimization. Examples in Computed Tomography and image deblurring illustrate the performance of the proposed method.