Robustness and stability of image-reconstruction algorithms have recently come under scrutiny. Their importance to medical imaging cannot be overstated. We review the known results for the topical variational regularization strategies ( ℓ2 and ℓ1 regularization) and present novel stability results for ℓp -regularized linear inverse problems for p∈(1,∞) . Our results guarantee Lipschitz continuity for small p and Hölder continuity for larger p . They generalize well to the Lp(Ω) function spaces.