This paper builds on classical distributionally robust optimization techniques to construct a comprehensive framework that can be used for solving inverse problems. Given an estimated distribution of inputs in X and outputs in Y, an ambiguity set is constructed by collecting all the perturbations that belong to a prescribed set K and are inside an entropy-regularized Wasserstein ball. By finding the worst-case reconstruction within K one can produce reconstructions that are robust with respect to various types of perturbations: X-robustness, Y|X-robustness and, more general, targeted robustness depending on noise type, imperfect forward operators and noise anisotropies. After defining the general robust optimization problem, we derive its (weak) dual formulation and we use it to design an efficient algorithm. Finally, we demonstrate the effectiveness of our general framework to solve matrix inversion and deconvolution problems defining K as the set of multivariate Gaussian perturbations in Y|X.