The objects of this paper are to introduce range-space variants of standard Krylov iterative solvers for unsymmetric and symmetric linear systems and to discuss how inexact matrix-vector products may be used in this context. The new range-space variants are characterized by possibly much lower storage and computational costs than their full-space counterparts, which is crucial in data assimilation applications and other inverse problems. However, this gain is achieved without sacrificing the inherent monotonicity properties of the original algorithms, which are of paramount importance in data assimilation applications. The use of inexact matrix-vector products is shown to further reduce computational cost in a controlled manner. Formal error bounds are derived on the size of the residuals obtained under two different accuracy models, and it is shown why a model controlling forward error on the product result is often preferable to one controlling backward error on the operator. Simple numerical examples finally illustrate the developed concepts and methods.