Inverse problems aim to recover a desired signal from noisy or corrupted measurements. These problems are inherently challenging due to the ill-posed nature of IPs, where solutions are not unique and are sensitive to noise. Classical methods for solving inverse problems typically involve minimizing a least-squares data fidelity term combined with a handcrafted regularization function. However, without careful design of the regularizer, such optimization approaches often yield suboptimal reconstructions, especially in general-purpose or ill-posed settings. Purely data-driven machine learning (ML) approaches have shown promising results by learning a direct mapping from measurements to ground-truth signals. While these methods often achieve superior reconstruction accuracy and faster runtime, they tend to lack interpretability and physical grounding. Model-based architectures such as loop unrolling draw inspiration from optimization by unrolling the iterative updates for an optimization-based solver and then learning a regularizer from data. These architectures are the current state-of-the-art ML-based inverse problem solvers, as they incorporate the forward model into each iteration to guide the reconstruction process. Despite their success, the robustness of ML-based approaches to inverse problems remains underexplored, and improving their robustness to noise and model mismatch continues to be an open research challenge. This thesis aims to enhance the robustness of model-based inverse problem solvers in two key dimensions: (1) robustness to measurement perturbations, under both fully known and partially specified forward models, and (2) robustness to forward model mismatch across supervised, semi-supervised, and test-time adaptation settings. The contributions of this thesis advance the development of more robust and more reliable machine-learning-based solvers for inverse problems, with significant implications for safety-critical or high-cost applications like medical imaging and seismic tomography.