Many methods of material characterization by rheological methods lead to ill-posed problems. The nature of ill-posedness is discussed. In this light, some such new problems appearing in molecular weight characterization of homopolymers as well as in the characterization of size distribution of dispersed phases in polymer blends by rheological techniques are analyzed. The working capability of a nonlinear regularization method is demonstrated with examples of blend theology. The role of Professor Meissner in motivating the research in the field of ill-posed problems as well as his important contribution to rheology of polymer blends is outlined.