In recent years, deep learning has achieved remarkable empirical success for image reconstruction. This has catalyzed an ongoing quest for the precise characterization of the correctness and reliability of data-driven methods in critical use cases, for instance, in medical imaging. Notwithstanding the excellent performance and efficacy of deep learning-based methods, concerns have been raised regarding the approaches' stability, or lack thereof, with serious practical implications. Significant advances have been made in recent years to unravel the inner workings of data-driven image recovery methods, challenging their widely perceived black-box nature. In this article, we specify relevant notions of convergence for data-driven image reconstruction, which forms the basis of a survey of learned methods with mathematically rigorous reconstruction guarantees. An example that is highlighted is the role of input-convex neural networks (ICNNs), offering the possibility to combine the power of deep learning with classical convex regularization theory for devising methods that are provably convergent. This survey article is aimed at both methodological researchers seeking to advance the frontiers of our understanding of data-driven image reconstruction methods as well as practitioners by providing an accessible description of useful convergence concepts and by placing some of the existing empirical practices on a solid mathematical foundation.