In many areas of applied sciences, engineering and technology there are three problems dealing with data and signals: (i) data compression; (ii) signal representations; and (iii) recovery of signals from partial or indirect information about the signals, often contaminated by noise. Major advances in these problems have been achieved in recent years where wavelets, multiresolution analysis, and kernel methods have played key roles. We consider problem (iii) and give an overview of specific contributions to inverse and ill-posed problems where reproducing kernel Hilbert spaces provide a natural setting.