After short introduction into polymer solution hydrodynamics, I shortly describe the main results on elastically driven instabilities and elastic turbulence in flow geometries with curvilinear streamlines, about prediction of elastic waves and absence of such waves in these flows. Then I switch to the main subject of the talk, namely recent development in parallel shear flows of polymer solutions proved to be linearly stable for the last about 50 years. We discovered in a straight channel flow elastically driven instability, elastic turbulence, elastic waves, and re-laminarization. In this regard, a short discussion of normal versus non-normal instabilities in such flows I give on the end.