  I shortly describe the main results on elastically driven instabilities and
elastic turbulence in viscoelastic inertia-less flows with curved streamlines.
Then I describe a theory of elastic turbulence and prediction of elastic waves
at Re<<1 and Wi>>1, which speed depends on the elastic stress similarly to the
Alfven waves in magneto-hydrodynamics and in in a contrast to all other fluid
flows with wave speed depending on medium elasticity. Since the established and
testified mechanism of elastic instability of viscoelastic flows with
curvilinear streamlines becomes ineffective at zero curvature, so parallel
shear flows are proved linearly stable, similar to Newtonian parallel shear
flows. However, the linear stability of parallel shear flows does not imply
their global stability. Here I switch to the main subject, namely a recent
development in inertia-less parallel shear channel flow of polymer solutions.
In such flow, we discover an elastically driven instability, elastic
turbulence, elastic waves, and drag reduction down to relaminarization that
contradicts the linear stability prediction. In this regard, I discuss shortly
normal versus non-normal bifurcations in such flows, flow resistance, velocity
and pressure fluctuations, and spatial and spectral velocity as a function of
Wi at a high elasticity number.
